Question 1

A 25 m long ladder is resting against a vertical wall. Initially, the foot of the ladder is 7 m away from the wall. If the top of the ladder then slides down the wall by 4 m, how far will its foot move away from the wall (in metres)?

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Introduction / Context: This problem combines the Pythagorean theorem with a dynamic situation: a ladder sliding down a wall. The ladder length remains constant, but its orientation and contact points change. We are asked to find how much the base of the ladder moves when the top descends by a known vertical distance. Given Data / Assumptions: Length of the ladder (hypotenuse) = 25 m. Initial horizontal distance from the wall (base) = 7 m. The wall is vertical and the ground is horizontal. The top of the ladder slides down by 4 m along the wall. The ladder does not slip at the wall or ground and remains straight. Concept / Approach: The ladder, wall and ground form a right triangle. The ladder is the hypotenuse, the vertical distance up the wall is one leg, and the horizontal distance from the wall is the other leg. Using the Pythagorean theorem both before and after the movement, we determine the initial and final horizontal distances. The difference between these two distances is the required movement of the foot. Step-by-Step Solution: Ladder length L = 25 m.Initial base distance b1 = 7 m.Initial height h1 satisfies: L^2 = h1^2 + b1^2.25^2 = h1^2 + 7^2 ⇒ 625 = h1^2 + 49.h1^2 = 625 − 49 = 576 ⇒ h1 = 24 m.The top slides down 4 m, so new height h2 = 24 − 4 = 20 m.Again, L^2 = h2^2 + b2^2.625 = 20^2 + b2^2 = 400 + b2^2.b2^2 = 625 − 400 = 225 ⇒ b2 = 15 m.Movement of the foot = b2 − b1 = 15 − 7 = 8 m. Verification / Alternative check: We can check quickly by verifying both triangles: (24, 7, 25) and (20, 15, 25) are Pythagorean triples since 24^2 + 7^2 = 576 + 49 = 625 and 20^2 + 15^2 = 400 + 225 = 625, both equal to 25^2. So the geometry is consistent and the base movement of 8 m is correct. Why Other Options Are Wrong: 6 m, 9 m, 10 m, and 5 m: All are arbitrary shifts that do not arise from correct Pythagorean calculations; they result from errors like subtracting heights directly or miscomputing the new base distance. Common Pitfalls: Students sometimes assume that the difference in heights equals the difference in base distances, which is incorrect. Others forget that the ladder length is constant and fail to apply the Pythagorean theorem properly a second time. Squaring errors (e.g., miscomputing 25^2 or 20^2) also commonly lead to incorrect results. Final Answer: The foot of the ladder moves away from the wall by 8 m.

Question 2

From a point 30 metres away from the foot of a flag post, the angles of elevation of the top and the bottom of a flag fixed on the post are 45° and 30° respectively. Assuming the post is vertical and the ground is level, what is the height of the fl…

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Introduction / Context: This problem distinguishes between the total height of a flag post and the height of the flag attached to it. Two different angles of elevation, to the top and the bottom of the flag, are observed from the same point, allowing us to calculate both heights and then subtract to obtain the flag's height. Given Data / Assumptions: Observer is 30 m horizontally from the base of the flag post. Angle of elevation to the top of the flag = 45°. Angle of elevation to the bottom of the flag = 30°. The post is vertical, and the ground is horizontal. We must find only the height of the flag (difference between these two vertical heights). Concept / Approach: Both lines of sight form right triangles with the same base of 30 m. The heights corresponding to 45° and 30° angles of elevation can be obtained via: tan(theta) = opposite / adjacentOnce we find the vertical height to the top and to the bottom of the flag, their difference equals the flag height. Step-by-Step Solution: Base distance d = 30 m.Height to top of flag, H_top satisfies: tan(45°) = H_top / 30.tan(45°) = 1 ⇒ H_top = 30 m.Height to bottom of flag, H_bottom satisfies: tan(30°) = H_bottom / 30.tan(30°) = 1 / √3 ⇒ H_bottom = 30 * (1 / √3) = 30 / √3.Rationalizing: 30 / √3 = 10√3.Using √3 ≈ 1.732 ⇒ H_bottom ≈ 10 * 1.732 = 17.32 m.Height of the flag = H_top − H_bottom ≈ 30 − 17.32 = 12.68 m. Verification / Alternative check: If the flag is approximately 12.68 m high, then adding its bottom height (~17.32 m) indeed gives the top at 30 m. Reversing the calculations back through tangent confirms that these heights correspond to angles of 30° and 45° respectively, verifying consistency. Why Other Options Are Wrong: 12√3 m (~20.78 m): This is larger than the total height difference and would place the top higher than 30 m.15 m and 14.32 m: Plausible but not matching the exact trigonometric calculations.10√3 m (~17.32 m): This is the height to the bottom of the flag, not the flag's height itself. Common Pitfalls: Many learners mistakenly compute the total height of the post or take only one of the heights instead of subtracting. Misapplying tan(30°) as √3 instead of 1 / √3 is another frequent source of error. Keeping track of which angle corresponds to which point (top or bottom of the flag) is critical. Final Answer: The height of the flag is approximately 12.68 m.

Question 3

A pilot in an aeroplane is flying at a constant altitude of 200 m above a river. He observes two points lying on opposite banks of the river such that the angles of depression to these two points are 45° and 60° respectively. Assuming the plane is v…

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Introduction / Context: This question involves angles of depression from a fixed altitude to two points on opposite sides of a river. By interpreting the angles as angles of elevation from the ground, we can form two right triangles sharing the same vertical height and find the horizontal distances to each bank. Adding these distances gives the width of the river. Given Data / Assumptions: The aeroplane flies at a constant height h = 200 m above the river. Angles of depression to the two points on opposite banks are 45° and 60°. The plane is vertically above a point between the two banks on a straight line perpendicular to the river. The river banks are assumed straight and parallel. Concept / Approach: Angles of depression from the horizontal are equal to the corresponding angles of elevation from the ground. Thus, for each bank we use: tan(theta) = opposite / adjacent = height / horizontal distanceWe then calculate the horizontal distance from the point directly beneath the plane to each bank and sum them to get the river width. Step-by-Step Solution: Let h = 200 m.For bank A with angle 45°: tan(45°) = h / d1 ⇒ 1 = 200 / d1 ⇒ d1 = 200 m.For bank B with angle 60°: tan(60°) = h / d2 ⇒ √3 = 200 / d2.So, d2 = 200 / √3 m.The two points lie on opposite sides, so the river width W = d1 + d2.Therefore, W = 200 + 200 / √3 m. Verification / Alternative check: We may approximate √3 ≈ 1.732. Then 200 / √3 ≈ 200 / 1.732 ≈ 115.47 m. The total width becomes approximately 200 + 115.47 = 315.47 m. Both triangles are consistent with the altitude of 200 m and the given angles using tan(45°) and tan(60°), confirming our formula. Why Other Options Are Wrong: (200 - 200/√3) m: This would correspond to an unrealistic subtraction of distances; width must be the sum of distances to opposite banks.400 √3 m and 400/√3 m: These are much larger or smaller than the correct width and do not match the individual triangle distances.200√3 m: Roughly 346 m, which would come from using tan(60°) with a single triangle rather than combining two banks properly. Common Pitfalls: Students commonly misinterpret “opposite banks” and subtract instead of add the distances. Another frequent mistake is reversing the roles of 45° and 60°, or misusing tan(60°) as 1 / √3. Remember that angles of depression equal angles of elevation, and that width must be the total separation between the two banks. Final Answer: The width of the river is (200 + 200/√3) m.

Question 4

A telegraph post is bent at some point above the ground so that its top just touches the ground at a distance of 8√3 metres from its foot and the broken top makes an angle of 30° with the horizontal. What is the original height of the post in metres?

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Introduction / Context: This is another broken-pole or broken-post type of height and distance question. The telegraph post was originally vertical, but after breaking at some point, its top touches the ground forming a right triangle. Using the given angle and the distance from its base, we can reconstruct both the remaining vertical part and the broken section to recover the original height. Given Data / Assumptions: The telegraph post is originally vertical on level ground. After breaking at some height, its top touches the ground at a horizontal distance of 8√3 m from the base. The broken part makes an angle of 30° with the ground. The broken part remains straight and attached to the stump. We must find the original total height of the post. Concept / Approach: The broken top is the hypotenuse of a right triangle with base 8√3 m and angle at the ground of 30°. From this, we find the length of the broken part using cos(30°). The vertical leg of this triangle is the height at which the post broke. The original height is then the sum of this vertical leg and the broken part, since that top piece was previously vertical as well. Step-by-Step Solution: Let L be the length of the broken part.Base (horizontal distance) = 8√3 m.Angle between broken part and ground = 30°.cos(30°) = base / hypotenuse = (8√3) / L.cos(30°) = √3 / 2, so √3 / 2 = (8√3) / L.Cancel √3 to get: 1 / 2 = 8 / L ⇒ L = 16 m.Vertical height of the breaking point = L * sin(30°).sin(30°) = 1 / 2 ⇒ height of break = 16 * 1 / 2 = 8 m.Original height of the post = vertical stump + broken part = 8 + 16 = 24 m. Verification / Alternative check: Check the right triangle with sides: hypotenuse 16 m, angle 30° at base, base 8√3 m, vertical leg 8 m. Using sin(30°) = opposite / hypotenuse gives 8 / 16 = 1 / 2, and cos(30°) = adjacent / hypotenuse gives (8√3) / 16 = √3 / 2, so the geometry is consistent and confirms the result. Why Other Options Are Wrong: 12 m, 16 m, 18 m, 20 m: Each is either the stump height, the hypotenuse length, or an arbitrary value. None equal stump + broken part, which must be 8 + 16. Common Pitfalls: One common error is to take only the vertical leg (8 m) or only the broken length (16 m) as the height of the post. Another is confusing which side is opposite and which is adjacent to the 30° angle, leading to using sine where cosine is required, or vice versa. Always sketch the triangle to avoid such mistakes. Final Answer: The original height of the telegraph post is 24 m.

Question 5

From two points on the same straight line as the base of a vertical tower, at horizontal distances of 4 m and 9 m from the foot of the tower, the angles of elevation of the top of the tower are complementary (their sum is 90°). What is the height of…

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Introduction / Context: This question uses the idea of complementary angles of elevation from two points at different distances from a tower. Complementary angles (adding to 90°) produce a relationship between tangent and cotangent, which we can exploit to find the tower's height without explicitly knowing the individual angles. Given Data / Assumptions: A vertical tower is on level ground. Two observation points lie along the same straight line as the tower base. Distances from the tower foot: 4 m (nearer point) and 9 m (farther point). Angles of elevation of the top of the tower from these two points are complementary (sum = 90°). We must find the height of the tower. Concept / Approach: If one angle of elevation is θ, the other is 90° − θ. For a height h and horizontal distance d, we have: tan(θ) = h / d1tan(90° − θ) = cot(θ) = h / d2Because tan(θ) * cot(θ) = 1, we can form an equation involving d1, d2 and h and solve for h. Step-by-Step Solution: Let h be the tower height.From the nearer point (distance 4 m): tan(θ) = h / 4.From the farther point (distance 9 m): tan(90° − θ) = cot(θ) = h / 9.We know tan(θ) * cot(θ) = 1.Thus: (h / 4) * (h / 9) = 1.h^2 / 36 = 1 ⇒ h^2 = 36.h = 6 m (positive height only). Verification / Alternative check: If h = 6 m, then from d1 = 4 m, tan(θ) = 6 / 4 = 1.5. From d2 = 9 m, tan(90° − θ) = h / 9 = 6 / 9 = 2 / 3. Since 1.5 and 2 / 3 are reciprocals, they are indeed tan(θ) and tan(90° − θ) respectively, confirming complementarity. Why Other Options Are Wrong: 4 m, 5 m, 7 m, 9 m: None of these values satisfy the product condition (h / 4) * (h / 9) = 1 when squared. They either make the product less than or greater than 1, violating the complementary angle property. Common Pitfalls: Students may assume specific angles like 30° and 60° without checking if they satisfy the distance ratios. Another error is adding or subtracting distances and heights instead of using the tangent relationships. Forgetting that tan and cot are reciprocals for complementary angles can also lead to incorrect setups. Final Answer: The height of the tower is 6 m.

Question 6

A helicopter is flying at an altitude of 1500 m above the sea. It observes two ships sailing in the same straight line and in the same direction. The angles of depression of the two ships from the helicopter are 60° and 30° respectively. What is the…

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Introduction / Context: This problem involves angles of depression from a helicopter to two ships at sea level. Since both ships are in the same direction from the helicopter and the helicopter's height is fixed, we can use trigonometry separately for each angle of depression and then find the distance between the ships along the sea surface. Given Data / Assumptions: Helicopter altitude h = 1500 m. Angles of depression to the nearer and farther ships are 60° and 30° respectively. The ships lie in the same straight line beneath the helicopter and in the same direction. Sea level is taken as a horizontal line, creating right triangles with the helicopter's vertical altitude. Concept / Approach: An angle of depression from the helicopter equals the angle of elevation from the ship. For each ship, we use: tan(theta) = opposite / adjacent = height / horizontal distanceWe calculate the horizontal distances d1 and d2 for 60° and 30°, and then take the difference d2 − d1 to find the separation between the ships. Step-by-Step Solution: Let h = 1500 m.For the nearer ship with angle 60°: tan(60°) = h / d1 ⇒ √3 = 1500 / d1.Thus, d1 = 1500 / √3 = 500√3 m.For the farther ship with angle 30°: tan(30°) = h / d2 ⇒ 1 / √3 = 1500 / d2.So, d2 = 1500√3 m.Distance between the two ships = d2 − d1 = 1500√3 − 500√3 = 1000√3 m. Verification / Alternative check: Approximating √3 ≈ 1.732, we get: d1 ≈ 500 * 1.732 ≈ 866 m,d2 ≈ 1500 * 1.732 ≈ 2598 m.Difference ≈ 2598 − 866 ≈ 1732 m, which equals 1000 * 1.732, confirming the expression 1000√3 m. Why Other Options Are Wrong: 1000/√3 m and 500/√3 m: Too small; they come from inverting the tangent relationships incorrectly.500√3 m: This is the distance from the helicopter's projection to the nearer ship, not the distance between the two ships.1500√3 m: This is the distance to the farther ship from the projection, again not the separation between the two ships. Common Pitfalls: Students sometimes add the distances instead of subtracting because they misinterpret the positions, or they mix up which angle corresponds to which distance. Another frequent mistake is to confuse tan(30°) with tan(60°) or to use 1 / tan instead of tan itself. Drawing a clear diagram helps avoid these issues. Final Answer: The distance between the two ships is 1000√3 m.

Question 7

From a point 30 m above the surface of a lake, the angle of elevation of an aeroplane in the sky is 30°, and the angle of depression of its image in the water is 60°. Assuming the water surface is perfectly horizontal and reflective, what is the hei…

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Introduction / Context: This interesting height-and-distance problem involves both an aeroplane and its mirror image in a lake. We use the geometry of reflection together with angles of elevation and depression from a point above the water to determine the plane's height above the water surface. Given Data / Assumptions: A point of observation is 30 m above the lake surface. Angle of elevation to the aeroplane from that point = 30°. Angle of depression to the plane's image in the lake = 60°. The plane is vertically above its image and the water surface acts as a perfect mirror. We need the height of the aeroplane from the water surface. Concept / Approach: Reflection in a horizontal mirror (the lake) places the image the same distance below the surface as the object is above it. Thus, if the aeroplane is H metres above the water, its image is H metres below the surface. From the observation point 30 m above the water, we can build two right triangles (to the plane and to its image) with a common horizontal distance. Using tangent relations for the two angles, we set up equations in H and solve. Step-by-Step Solution: Let H be the height of the plane above the water surface.Place the water surface at y = 0, the observation point O at y = 30, and the plane at y = H.The image of the plane is at y = −H.Let the horizontal distance from O to the vertical line through the plane be x.Angle of elevation to the plane = 30°:tan(30°) = (H − 30) / x = 1 / √3.Thus, x = √3(H − 30).Angle of depression to the image = 60°:tan(60°) = (30 + H) / x = √3.So, x = (30 + H) / √3.Equate the two expressions for x:√3(H − 30) = (30 + H) / √3.Multiply both sides by √3: 3(H − 30) = 30 + H.3H − 90 = 30 + H ⇒ 2H = 120 ⇒ H = 60 m. Verification / Alternative check: If H = 60 m, then: H − 30 = 30 m and H + 30 = 90 m.Using x = √3(H − 30) = 30√3 m, tan(30°) = 30 / (30√3) = 1 / √3, correct.Also, tan(60°) = 90 / (30√3) = 90 / (51.96...) ≈ √3, which fits well. Why Other Options Are Wrong: 45 m, 50 m, 75 m, 90 m: Each of these values fails to satisfy both tangent equations simultaneously. Substituting them will yield inconsistent horizontal distances x for the two angles. Common Pitfalls: Beware of forgetting that the image is the same distance below the water as the plane is above it, leading to wrong vertical differences. Some students also confuse angles of elevation and depression or assume H − 30 and H + 30 incorrectly. Always draw a diagram marking heights carefully. Final Answer: The height of the aeroplane above the water surface is 60 m.

Question 8

The angles of elevation of the top of a 220 m high tree from two points on the same level ground are 30° and 45°. What is the distance in metres between these two points of observation?

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Introduction / Context: This problem is similar to others involving heights observed from two different locations. Here, the tree height is known, and the angles of elevation from two points are given. We must find the horizontal distance between these observation points by finding each distance from the tree and then subtracting. Given Data / Assumptions: Height of the tree = 220 m. Angle of elevation to the top from one point = 30°. Angle of elevation to the top from the other point = 45°. Both points lie on the same straight line on level ground with the foot of the tree. We are asked for the distance between the two observation points. Concept / Approach: For a vertical tree of height h and horizontal distance d from a point, we use: tan(theta) = h / d ⇒ d = h / tan(theta)The point with the smaller angle is farther from the tree. After computing d30 and d45, the distance between the points is |d30 − d45|. Step-by-Step Solution: Let h = 220 m.For the 30° observation: tan(30°) = 1 / √3.Thus, distance d30 = h / tan(30°) = 220 / (1 / √3) = 220√3 m.For the 45° observation: tan(45°) = 1.So, distance d45 = h / tan(45°) = 220 / 1 = 220 m.Distance between the two points = d30 − d45 = 220√3 − 220.Factor: = 220(√3 − 1) m.Approximating √3 ≈ 1.732 ⇒ √3 − 1 ≈ 0.732.So, distance ≈ 220 * 0.732 ≈ 161.04 m, usually rounded to 161.05 m. Verification / Alternative check: Taking d30 ≈ 220 * 1.732 ≈ 381.04 m, and d45 = 220 m, we get their difference as 161.04 m, consistent with 161.05 m to two decimal places. This confirms our symbolic expression as well as the numerical option provided. Why Other Options Are Wrong: 193.22 m, 144.04 m, 176.12 m: These are alternate distances that do not correspond to 220(√3 − 1). Substituting them back does not give consistent tangent values.220(√3 - 1) m is the exact symbolic value; numerically it matches approximately 161.05 m, not any of the other decimals listed. Common Pitfalls: Some learners mistakenly add the distances instead of subtracting or mix up which angle corresponds to which distance (bigger angle means closer point). Others miscalculate tan(30°) or forget to use the reciprocal when moving from tan(30°) to distance. Using an accurate approximation for √3 is important for numerical answers. Final Answer: The distance between the two observation points is approximately 161.05 m.

Question 9

The angles of elevation of the top of a tower 72 metres high, as seen from the top and from the bottom of a nearby building on the same horizontal line, are 30° and 60° respectively. What is the height of the building in metres?

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Introduction / Context: This question involves two vertical structures: a known tower and an unknown building. The angles of elevation of the tower top from the top and bottom of the building are given. Using right-triangle trigonometry, we can set up two equations and solve for both the horizontal distance and the building height, then extract the required height. Given Data / Assumptions: Height of the tower = 72 m. Angle of elevation from the top of the building = 30°. Angle of elevation from the bottom of the building = 60°. The tower and building stand on the same level ground. The points (tower base, building base, building top) lie along one straight horizontal line. Concept / Approach: Let the building height be h and the horizontal distance between the tower and the building be x. From the bottom of the building, the tower top is at height 72 m, giving tan(60°) = 72 / x. From the top of the building, the vertical difference between the two tops is (72 − h), giving tan(30°) = (72 − h) / x. By solving these two equations simultaneously, we find h. Step-by-Step Solution: Let h be the building height and x the horizontal distance.From the building bottom: tan(60°) = 72 / x.tan(60°) = √3 ⇒ √3 = 72 / x ⇒ x = 72 / √3 = 24√3.From the building top: tan(30°) = (72 − h) / x.tan(30°) = 1 / √3 ⇒ 1 / √3 = (72 − h) / x.Substitute x = 24√3: 1 / √3 = (72 − h) / (24√3).Cross-multiply: (72 − h) = (24√3) / √3 = 24.Therefore, h = 72 − 24 = 48 m. Verification / Alternative check: With h = 48 m and x = 24√3, from the bottom: tan(60°) = 72 / (24√3) = 3 / √3 = √3, correct. From the top: the difference in height is 72 − 48 = 24 m, so tan(30°) = 24 / (24√3) = 1 / √3, also correct. Both triangles check out perfectly. Why Other Options Are Wrong: 42 m and 36 m: These intermediate values do not satisfy both tangent equations simultaneously.20√3 m and 24√3 m: These are associated with horizontal distances, not the building height itself, although 24√3 does appear in the working as x. Common Pitfalls: One common mistake is to assume the building is taller than the tower or misinterpret which height difference to use from the top of the building. Another is mixing up tan(30°) and tan(60°). Carefully identify vertical differences (72 − h) from the top and full height 72 from the bottom to build correct equations. Final Answer: The height of the building is 48 m.

Question 10

A balloon rises vertically from a point P on level ground at a uniform speed. After 6 minutes, an observer standing at a point on the ground that is 450√3 metres away from P measures the angle of elevation of the balloon as 60°. Assuming the observa…

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Introduction / Context: This problem illustrates vertical motion combined with an oblique observation from a point on the ground. The balloon rises straight up from its starting point, while the observer stands some horizontal distance away. Using the angle of elevation after a given time, we compute the vertical height and hence the speed of ascent in metres per second. Given Data / Assumptions: The balloon starts from point P on the ground. An observer is at a point Q, 450√3 m horizontally from P. After 6 minutes, the angle of elevation of the balloon from Q is 60°. The balloon rises vertically with uniform speed. Ground between P and Q is level. Concept / Approach: The balloon's position after 6 minutes forms a right triangle with vertical leg equal to the balloon height and horizontal leg equal to the fixed distance PQ. The angle between PQ and the line of sight is the angle of elevation. Using tan(theta) = opposite / adjacent, we find the height reached in 6 minutes and then divide by the time in seconds to find the speed in m/s. Step-by-Step Solution: Horizontal distance PQ = 450√3 m.Angle of elevation after 6 minutes = 60°.Let h be the height of the balloon at that time.Then tan(60°) = h / (450√3).tan(60°) = √3 ⇒ √3 = h / (450√3).So, h = √3 * 450√3 = 450 * 3 = 1350 m.Time taken = 6 minutes = 6 * 60 = 360 seconds.Speed v = distance / time = h / t = 1350 / 360 m/s.Simplify: 1350 / 360 = 135 / 36 = 15 / 4 = 3.75 m/s. Verification / Alternative check: We can check tan(60°) from the computed height. Using h = 1350 m and base = 450√3 m, tan(60°) = 1350 / (450√3) = 3 / √3 = √3, as required. This confirms that our triangle is consistent with the given angle. Why Other Options Are Wrong: 4.25 m/s, 4.5 m/s, 3.45 m/s, 5 m/s: These speeds correspond to different heights over the given 360 seconds and would not yield tan(60°) = √3 when combined with the fixed base of 450√3 m. Common Pitfalls: Common mistakes include forgetting to convert minutes to seconds, mixing up which side is opposite and which is adjacent to the angle, or incorrectly simplifying tan(60°). Some learners also mistakenly treat 450√3 as the hypotenuse rather than the base, which leads to incorrect height and speed calculations. Final Answer: The speed of the balloon is 3.75 m/s.

Question 11

From a point on level ground, the angle of elevation of the top of a building is 45°. After a person walks 100 metres straight towards the building in the same horizontal line, the angle of elevation becomes 60°. What is the height of the building (…

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Introduction / Context: This question involves a person moving closer to a building, causing the angle of elevation of the top to increase from 45° to 60°. Using trigonometry and the known horizontal movement, we can form equations to find the horizontal distance to the building and hence its height. Given Data / Assumptions: Initial angle of elevation to the top of the building = 45°. After walking 100 m towards the building, the angle of elevation = 60°. The walking path is along a straight horizontal line directly towards the building. The building stands vertically on level ground. We are asked to find the building height. Concept / Approach: Let x be the initial horizontal distance from the observer to the building and h the building height. Initially, tan(45°) = h / x, giving h = x. After walking 100 m, the new distance is (x − 100), and tan(60°) = h / (x − 100). Substituting h = x into the second equation allows us to solve for x and then find h. Step-by-Step Solution: Let h be the height of the building and x the initial distance.From the first position: tan(45°) = h / x ⇒ 1 = h / x ⇒ h = x.From the second position: distance to building = x − 100.tan(60°) = h / (x − 100) ⇒ √3 = h / (x − 100).Substitute h = x: √3 = x / (x − 100).Cross-multiply: √3(x − 100) = x.√3x − 100√3 = x ⇒ √3x − x = 100√3.x(√3 − 1) = 100√3 ⇒ x = 100√3 / (√3 − 1).Multiply numerator and denominator by (√3 + 1):x = 100√3(√3 + 1) / (3 − 1) = 50√3(√3 + 1).Since √3(√3 + 1) = 3 + √3, we get x = 50(3 + √3).But h = x, so height h = 50(3 + √3) m. Verification / Alternative check: Approximating √3 ≈ 1.732, h ≈ 50(3 + 1.732) = 50 * 4.732 ≈ 236.6 m. Initially, distance x ≈ 236.6 m gives tan(45°) = h / x ≈ 1. After walking 100 m, new distance ≈ 136.6 m, and h / (x − 100) ≈ 236.6 / 136.6 ≈ 1.732 = √3, which is tan(60°). This confirms consistency. Why Other Options Are Wrong: 100(√3 + 1) m and 50(√3 + 1) m: These come from partial or incorrect algebraic manipulation and do not satisfy both angle conditions.150 m and 100√3 m: These simple values fail when substituted back into the tangent relations for both 45° and 60°. Common Pitfalls: Students often forget that the building height equals the initial distance when the angle is 45°, or they mistakenly use (x + 100) instead of (x − 100) for the second distance. Confusion between tan(45°) and tan(60°), as well as arithmetic errors in handling √3, are also common. Final Answer: The height of the building is 50(3 + √3) m.

Question 12

The upper part of a tree is broken by the wind so that the broken top makes an angle of 60° with the ground and touches the ground at a point 25 metres from the foot of the tree. What was the original height of the tree (in metres) before it broke?

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Introduction / Context: This is another broken-tree trigonometry question. The upper part of the tree bends over and touches the ground, forming a right triangle with the ground and the remaining upright part of the trunk. We use the given angle and distance along the ground to compute the original height of the tree. Given Data / Assumptions: The tree was initially vertical. The top breaks and touches the ground at a point 25 m from the base. The broken top makes an angle of 60° with the ground. The broken part remains straight and attached at the breaking point. We must find the total original height of the tree. Concept / Approach: The broken upper part is the hypotenuse of a right triangle whose base is 25 m and angle at the ground is 60°. From this, we compute the length of the broken part (hypotenuse) using cos(60°). The vertical projection of this hypotenuse is the height at which the tree broke. Adding this break height to the hypotenuse length returns the original height of the tree, because before breaking that top piece stood vertically above the break point. Step-by-Step Solution: Let L be the length of the broken top of the tree.Base distance from tree foot to top's contact point = 25 m.Angle between broken part and ground = 60°.cos(60°) = base / hypotenuse = 25 / L.cos(60°) = 1 / 2 ⇒ 1 / 2 = 25 / L ⇒ L = 25 * 2 = 50 m.Vertical height from ground to point of break = L * sin(60°).sin(60°) = √3 / 2 ⇒ break height = 50 * (√3 / 2) = 25√3 m.Original height H = break height + length of broken part = 25√3 + 50.Approximate: √3 ≈ 1.732 ⇒ 25√3 ≈ 25 * 1.732 = 43.3.So H ≈ 43.3 + 50 = 93.3 m. Verification / Alternative check: We can verify by reconstructing the triangle: if the hypotenuse is 50 m and angle 60°, the base should be 50 * cos(60°) = 25 m, matching the given base distance. The vertical leg is 50 * sin(60°) = 25√3, so adding this to the hypotenuse gives the original height. Both the geometry and the arithmetic are consistent. Why Other Options Are Wrong: 84.14 m, 98.25 m, 120.24 m, 75 m: These values do not correspond to 25√3 + 50 and do not satisfy the right-triangle relations with base 25 m and angle 60°. Plugging any of them back fails the trigonometric relationships. Common Pitfalls: Common mistakes include treating 25 m as the length of the broken top instead of the horizontal base, or forgetting to add the stump height and the broken part. Others misapply sin(60°) and cos(60°), or assume that the original height equals only one of the legs. A quick diagram and correct identification of the hypotenuse make the process much clearer. Final Answer: The original height of the tree was approximately 93.3 m.

Question 13

One tower has a height of 300 metres. From the top of another tower standing on the same level ground and 120 metres away horizontally, the angle of depression of the top of the 300 m tower is 60°. What is the height of this smaller tower (in metres…

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Introduction / Context: This height-and-distance question involves two towers of different heights. From the top of one tower, the top of the other is seen below the horizontal line of sight, giving an angle of depression. By relating the vertical difference in heights to the horizontal separation, we can use the tangent function to find the unknown height. Given Data / Assumptions: Tower A has a known height of 300 m. Tower B has unknown height h and is referred to as the smaller tower. Horizontal distance between the bases of the two towers = 120 m. From the top of Tower A, the angle of depression of the top of Tower B is 60°. The ground between the towers is level. Concept / Approach: An angle of depression from the horizontal line at the top of Tower A down to the top of Tower B corresponds to an angle of elevation of 60° from Tower B up to that horizontal line. The vertical difference in height between the two towers is (300 − h). Using the right triangle formed by this vertical difference and the horizontal distance, we apply: tan(60°) = (vertical difference) / (horizontal distance)and solve for h. Step-by-Step Solution: Let h be the height of the smaller tower (Tower B).Vertical difference between tops = 300 − h.Horizontal distance between towers = 120 m.Given angle of depression = 60°, so:tan(60°) = (300 − h) / 120.We know tan(60°) = √3.Thus: √3 = (300 − h) / 120.So, 300 − h = 120√3.Therefore, h = 300 − 120√3.Using √3 ≈ 1.732: 120√3 ≈ 120 * 1.732 = 207.84.Hence, h ≈ 300 − 207.84 = 92.16 m, which matches the option 92.15 m when rounded. Verification / Alternative check: We can quickly verify: vertical difference ≈ 207.84 m and horizontal distance 120 m, so tan(60°) ≈ 207.84 / 120 ≈ 1.732, which equals √3. This agrees with the given angle of depression and confirms the correctness of the height found. Why Other Options Are Wrong: 88.24 m, 106.71 m, 112.64 m, 120 m: These heights lead to vertical differences that do not satisfy (300 − h) / 120 = √3. Substituting any of them will produce tangent values different from tan(60°). Common Pitfalls: Some students mistakenly compute h = 300 + 120√3, making the smaller tower taller than the 300 m one. Others confuse angle of elevation and depression or forget that the angle is formed with the horizontal, not with the vertical. Always draw a diagram and mark the vertical difference and horizontal distance clearly to avoid sign and interpretation errors. Final Answer: The height of the smaller tower is approximately 92.15 m.

Question 14

Two trees stand on opposite sides of a straight road. The distance between the two trees is 400 metres. A point P lies on the road between them. The angles of depression of point P from the tops of the two trees are 45° and 60°. If the height of the…

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Introduction / Context: This problem is a classic height and distance question from trigonometry. We are given angles of depression from the tops of two trees to a point on a road between them, and the height of one tree. Using basic trigonometric ratios, we can find the height of the second tree. Given Data / Assumptions: Two trees stand on opposite sides of a straight road. Distance between the trees along the road = 400 metres. A point P lies on the road between the trees. Angle of depression to P from the top of the first tree = 45°. Height of this first tree = 200 metres. Angle of depression to P from the top of the second tree = 60°. The road is assumed to be horizontal and the trees vertical. Concept / Approach: Angles of depression from the top of a vertical object to a point on the ground correspond to angles of elevation from that ground point back to the top. In a right triangle, tan(θ) = opposite / adjacent, where the opposite side is the height and the adjacent side is the horizontal distance from the point on the ground to the foot of the tree. We will set up two such right triangles, use tangent values for 45° and 60°, and use the fact that the point lies between the two trees so that the two horizontal distances add up to 400 metres. Step-by-Step Solution: Let the trees be at points A and B, with A on the left and B on the right, and let P be the point on the road between them. Let AP be the horizontal distance from the foot of the 200 metre tree to point P. For the first tree, tan 45° = height / AP = 200 / AP. Since tan 45° = 1, we get 1 = 200 / AP, so AP = 200 metres. The total distance between the trees is AB = 400 metres, so the remaining distance BP = AB - AP = 400 - 200 = 200 metres. Let the height of the second tree be h metres. For the second tree, tan 60° = h / BP = h / 200. We know tan 60° = √3, so √3 = h / 200, which gives h = 200√3 metres. Verification / Alternative check: We can quickly check reasonableness. The point P is equally distant horizontally from both trees (200 metres each side). Since the angle of depression from the taller tree is 60°, which is larger than 45°, the taller tree must have greater height. Using h = 200√3 gives h approximately equal to 200 × 1.732 = 346.4 metres, which is indeed taller than 200 metres and consistent with a larger angle. Why Other Options Are Wrong: 200: This would mean both trees have the same height, which cannot produce different angles of depression from a symmetric position on the road. 100√3: This is less than 200, so it would give a smaller angle of depression than 45°, not 60°. 250: This height is larger than 200 but does not satisfy tan 60° = height / 200, so it does not match the trigonometric condition. Common Pitfalls: Students often mix up angles of elevation and depression or forget that the point lies between the trees, so the two horizontal distances must sum to 400 metres. Another common mistake is to use sine instead of tangent, or to incorrectly take the distance between trees as one of the legs of the right triangle without splitting it at the point P. Final Answer: Thus, the height of the other tree is 200√3 metres.

Question 15

From a point P on the ground, the angle of elevation of the top of a vertical tower is such that tan θ = 3/4. After walking 560 metres straight towards the foot of the tower, the tangent of the angle of elevation of the tower becomes 4/3. What is th…

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Introduction / Context: This is a height and distance problem involving a tower and two different positions of an observer. The key feature is that the tangent of the angle of elevation changes when the observer walks closer to the tower. We can use the definition of tangent in right triangles to set up equations and solve for the height of the tower. Given Data / Assumptions: At the first point P, tan θ₁ = 3/4 for the angle of elevation to the top of the tower. The observer walks 560 metres towards the tower in a straight line. At the new position, tan θ₂ = 4/3 for the angle of elevation. The tower is vertical and the ground is taken to be horizontal. We are required to find the height of the tower in metres. Concept / Approach: Let the horizontal distance from the initial point P to the foot of the tower be x metres, and let the height of the tower be h metres. At each position, the tangent of the angle of elevation is given by tan θ = opposite side (height) divided by adjacent side (horizontal distance). We will use tan θ₁ = 3/4 and tan θ₂ = 4/3 to form two equations in h and x, then solve for h. Step-by-Step Solution: Let x be the initial horizontal distance from P to the foot of the tower. Let h be the height of the tower. At the first position, tan θ₁ = h / x = 3 / 4, so h = (3x) / 4. After walking 560 metres towards the tower, the new distance from the tower is x - 560 metres. At the second position, tan θ₂ = h / (x - 560) = 4 / 3. Substitute h from the first equation: (3x / 4) / (x - 560) = 4 / 3. This gives 3x / (4x - 2240) = 4 / 3. Cross multiply: 3 × 3x = 4 × (4x - 2240). 9x = 16x - 8960, so 7x = 8960 and x = 1280 metres. Now h = (3x) / 4 = (3 × 1280) / 4 = 3840 / 4 = 960 metres. Verification / Alternative check: Check the tangents with h = 960 and x = 1280. At the first point, h / x = 960 / 1280 = 3 / 4, which matches tan θ₁. At the second point, horizontal distance is 1280 - 560 = 720, and h / 720 = 960 / 720 = 4 / 3, which matches tan θ₂. So the value is consistent. Why Other Options Are Wrong: 720: This would give tan θ₁ = 720 / 1280 = 9 / 16, which does not match 3 / 4. 840: Using this height does not satisfy both tangent conditions simultaneously. 1030: This also fails to satisfy the exact fraction values 3 / 4 and 4 / 3 for the given 560 metre movement. Common Pitfalls: Common errors include using the wrong sign for the change in distance, forgetting that the second horizontal distance is x - 560 (not x + 560), or trying to use sine or cosine instead of tangent. Another mistake is attempting to plug approximate decimal tangent values instead of using the given simple fractions, which can create unnecessary rounding issues. Final Answer: The height of the tower is 960 metres.

Question 16

An object travels at a constant speed of 5 feet per second. How many feet will it travel in exactly one hour at this speed?

Accepted Answer

Introduction / Context: This problem is a straightforward application of the relationship between speed, time, and distance. Many aptitude and school level questions involve such unit conversions, especially when changing from seconds to minutes or hours. Correctly handling units is crucial because a small mistake in unit conversion can dramatically change the final answer, even when the arithmetic itself is simple. Given Data / Assumptions: Speed of the object = 5 feet per second. Time of travel = 1 hour. We need to find the total distance traveled in feet. The speed is constant throughout the hour. Concept / Approach: The basic formula connecting distance, speed, and time is: distance = speed * time. Here, speed is given in feet per second, while time is specified in hours. To use the formula correctly, both must be in compatible units. We can either convert speed to feet per hour or convert time to seconds. Because 1 hour equals 60 minutes and each minute has 60 seconds, 1 hour equals 3600 seconds. Once we know the total number of seconds in one hour, we multiply by the speed in feet per second to get the total distance in feet. Step-by-Step Solution: Step 1: Write the formula. distance = speed * time. Step 2: Convert time from hours to seconds. 1 hour = 60 minutes. Each minute = 60 seconds. So, 1 hour = 60 * 60 = 3600 seconds. Step 3: Multiply speed by time in seconds. Speed = 5 feet per second. Time = 3600 seconds. distance = 5 * 3600 = 18000 feet. Therefore, the object travels 18000 feet in one hour. Verification / Alternative check: Another way to think about this is to first find the distance covered in one minute. At 5 feet per second, in one minute there are 60 seconds, so distance per minute is 5 * 60 = 300 feet. Then, in one hour, which has 60 minutes, the distance is 300 * 60 = 18000 feet. This alternative method leads to the same result and checks the correctness of the earlier calculation. Why Other Options Are Wrong: 3000: This would correspond to a much shorter time period or a lower speed; it is far less than the correct distance for one full hour at 5 feet per second. 1800: This is the distance traveled in 6 minutes at 5 feet per second (5 * 60 * 6), not in one hour. 15000: This could result from mixing up conversions or using 50 minutes instead of 60; it does not match the correct product 5 * 3600. 3600: This is the number of seconds in an hour, not the distance in feet. Confusing time and distance is a common unit error. Common Pitfalls: Many students forget to convert hours into seconds and mistakenly multiply 5 by 60 or 5 by 600. Others compute 5 * 60 and stop, assuming that is enough. Keeping careful track of units and knowing that one hour equals 3600 seconds are essential. It helps to write out all conversion steps explicitly before performing multiplications. Final Answer: The object travels 18000 feet in one hour.

Question 17

An observer who is 1.6 m tall is standing at a horizontal distance of 20√3 m from the base of a vertical tower. The angle of elevation from the observer's eye level to the top of the tower is 30°. What is the height of the tower?

Accepted Answer

Introduction / Context: This question is a classic example of a height and distance problem involving trigonometry. It uses the concept of angle of elevation and right triangles to determine the height of a tower observed from a certain distance. Such problems commonly appear in aptitude exams, and they test your ability to apply trigonometric ratios, especially tangent, in real life style situations. Accounting for the observer's own height is an important part of accurately computing the total height of the tower. Given Data / Assumptions: Height of the observer (eye level) = 1.6 m. Horizontal distance from observer to the base of the tower = 20√3 m. Angle of elevation from the observer's eye to the top of the tower = 30°. The tower is vertical and the ground is assumed to be horizontal. We need to find the total height of the tower from the ground. Concept / Approach: In a right triangle formed by the tower, the ground, and the line of sight, the tangent of the angle of elevation is given by: tan(theta) = opposite side / adjacent side. Here, the opposite side is the vertical distance from the observer's eye to the top of the tower, and the adjacent side is the horizontal distance from the observer to the tower. Using the given angle of 30°, and the identity tan(30°) = 1 / √3, we can first find the vertical height above the observer's eye. Then we add the observer's height (1.6 m) to obtain the total height of the tower. Step-by-Step Solution: Step 1: Set up the right triangle. Let h be the height of the tower. Height from observer's eye to top of tower = h - 1.6. Horizontal distance from observer to tower = 20√3 m. Step 2: Use the tangent ratio. tan(30°) = (h - 1.6) / (20√3). Step 3: Substitute tan(30°) = 1 / √3. 1 / √3 = (h - 1.6) / (20√3). Step 4: Cross multiply to solve for h - 1.6. h - 1.6 = (20√3) * (1 / √3) = 20. Step 5: Add the observer's height to find total tower height. h = 20 + 1.6 = 21.6 m. Therefore, the height of the tower is 21.6 m. Verification / Alternative check: We can verify the result by plugging back into the tangent formula. If h = 21.6 m, then the vertical distance from eye to top is 21.6 - 1.6 = 20 m. The horizontal distance is 20√3 m. So tan(30°) = opposite / adjacent = 20 / (20√3) = 1 / √3, which matches the known value of tan(30°). This confirms that our calculated tower height is consistent with the trigonometric relationship and the given data. Why Other Options Are Wrong: 23.2 m and 24.72 m: These values produce a vertical difference that does not match tan(30°) when combined with the horizontal distance of 20√3 m, so the trigonometric relation fails. 19.6 m: This height is less than the vertical distance implied by the angle and distance; substituting it back into tan(30°) does not yield 1 / √3. None of these: This is incorrect because we have found a specific option, 21.6 m, that satisfies all conditions exactly. Common Pitfalls: Some students forget to add the observer's height after calculating the vertical distance using trigonometry, thus giving only the height above eye level instead of the full tower height. Others may mix up sine, cosine, and tangent, or misremember that tan(30°) equals 1 / √3. Ensuring a clear diagram, carefully identifying opposite and adjacent sides, and remembering the correct trigonometric values prevents most of these mistakes. Final Answer: The height of the tower is 21.6 m.

Question 18

A man is standing at a point P and observes the top of a vertical tower. The angle of elevation from point P to the top of the tower is 30°. He then walks some distance straight towards the tower, and now the angle of elevation to the top becomes 60…

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Introduction / Context: This question deals with height and distance using trigonometry, where a man observes a tower from two different positions. The angles of elevation change, but the actual distances walked are not specified. Such questions are meant to test not only trigonometric skills but also your understanding of when information is sufficient or insufficient to determine a unique numerical answer. Recognizing data inadequacy is as important as performing calculations correctly. Given Data / Assumptions: The tower is vertical and stands on level ground. From point P, the angle of elevation to the top of the tower is 30°. The man walks some distance directly towards the tower. From the new point, the angle of elevation to the top becomes 60°. No numeric value of the walking distance or the height of the tower is given. We are asked for the distance between the base of the tower and the original point P. Concept / Approach: If we let x be the initial distance from P to the tower base and h be the height of the tower, then using basic trigonometry: tan(30°) = h / x and tan(60°) = h / (x - d), where d is the distance walked towards the tower. These equations give relationships between h, x, and d. However, unless we are given an actual numeric value for h or d, or a further condition, we will not be able to solve for x uniquely. The system of equations only provides ratios, not absolute measures. Step-by-Step Reasoning: Step 1: Set up variables. Let the height of the tower be h. Let the initial distance from P to the tower be x. Let the man walk a distance d towards the tower, so the new distance is x - d. Step 2: Use the first angle of elevation. tan(30°) = h / x. Since tan(30°) = 1 / √3, we get h = x / √3. Step 3: Use the second angle of elevation. tan(60°) = h / (x - d). tan(60°) = √3, so h = √3 (x - d). Step 4: Equate the two expressions for h. x / √3 = √3 (x - d). Multiply both sides by √3: x = 3(x - d) = 3x - 3d. Step 5: Rearrange to relate x and d. x = 3x - 3d implies 2x = 3d, so d = (2/3)x. We now only know that d is two thirds of x, but we do not know their actual numeric values. Verification / Alternative check: The equations derived show a proportional relationship between x and d but do not fix their magnitudes. For example, if x were 3 units, then d would be 2 units; if x were 30 units, then d would be 20 units. In both cases, the angles of 30° and 60° could be satisfied with different tower heights. Thus, there is an infinite family of solutions that satisfy the trigonometric conditions, and no unique numeric value of x can be determined from the given information alone. Why Other Options Are Wrong: 8 units and 12 units: These are specific distances that cannot be justified because the equations do not yield a single numeric value for x. Without additional constraints, choosing any particular distance is arbitrary. None of these: This is less precise because we actually have a clear reason that the data is insufficient; the correct description is that the data is inadequate or that we cannot determine a unique numeric answer. Cannot be determined exactly: This is essentially similar in meaning to data inadequate, but the standard exam phrasing here is best captured by the option explicitly stating data inadequacy. Common Pitfalls: Many students try to assume a specific value for the distance walked or the height of the tower, even though the question does not provide such information. This leads to arbitrary calculations and incorrect answers. Another pitfall is to assume that two equations always guarantee a unique solution, without checking how many unknowns are involved. In this case, there are three unknowns (h, x, d) but only two independent equations, so one degree of freedom remains, preventing a unique answer for x. Final Answer: The information given is not sufficient; the distance from the base of the tower to point P is data inadequate.

Question 19

A vertical toy that is 18 cm long casts a shadow 8 cm long on level ground at a certain time of day. At the same time and in the same light, a pole casts a shadow 48 m long on the ground. Assuming both the toy and the pole are vertical, what is the …

Accepted Answer

Introduction / Context: This question is a classic application of similar triangles in height and distance problems. When the sun's rays fall at the same angle, any vertical objects and their shadows form similar right triangles. This means that the ratio of height to shadow length is the same for all such objects at that moment. Knowing the dimensions of a small object and its shadow allows us to scale up to find the height of a larger object from its shadow length. Given Data / Assumptions: Height of the toy = 18 cm. Length of the toy's shadow = 8 cm. Length of the pole's shadow = 48 m. Both toy and pole are vertical and stand on level ground. Sun angle is the same for both, so the triangles formed are similar. We need to find the height of the pole. Concept / Approach: When two right triangles are similar, corresponding sides are proportional. Here, the toy and its shadow form a smaller right triangle, and the pole and its shadow form a larger, similar triangle. The ratio of height to shadow length for the toy equals the ratio of height to shadow length for the pole. We must also be careful with units: the toy measurements are in centimeters, while the pole's shadow is given in meters. It is helpful to work in consistent units, either all in centimeters or all in meters. Step-by-Step Solution: Step 1: Express the ratio of height to shadow for the toy. Toy height = 18 cm, toy shadow = 8 cm. Ratio = 18 / 8 = 9 / 4. Step 2: Convert the pole's shadow length to a compatible unit. Pole shadow = 48 m. Since 1 m = 100 cm, 48 m = 4800 cm. Step 3: Use similarity of triangles. Let H be the height of the pole in centimeters. Then H / 4800 = 18 / 8 = 9 / 4. Step 4: Solve for H. H = 4800 * (9 / 4) = 4800 * 9 / 4. First, 4800 / 4 = 1200. Then 1200 * 9 = 10800 cm. Step 5: Convert the pole height back to meters. 10800 cm = 10800 / 100 m = 108 m. Therefore, the height of the pole is 108 m. Verification / Alternative check: We can quickly check the proportionality. For the toy, height to shadow ratio is 18 : 8 = 9 : 4. For the pole, with height 108 m and shadow 48 m, the ratio is 108 : 48. Simplify by dividing both by 12: 108 / 12 = 9 and 48 / 12 = 4, so we also get 9 : 4. Because both ratios match, the triangles are indeed similar, and our scaled height is consistent with the geometry of the problem. This confirms that 108 m is correct. Why Other Options Are Wrong: 1080 cm: This equals 10.8 m, which is ten times smaller than the correct height and does not maintain the 9 : 4 ratio with a 48 m shadow. 180 m: With a 48 m shadow, this would give a ratio of 180 : 48, which simplifies to 15 : 4, not 9 : 4, so it is inconsistent with the toy's ratio. 118 cm: This is even shorter than the toy in terms of centimeters and clearly does not scale correctly to a 48 m shadow, so this option is unrealistic here. 10.8 m: This is the same as 1080 cm, again too small and does not preserve the ratio of similar triangles. It seems to result from an incorrect scaling by a factor of 10 instead of 100. Common Pitfalls: The most common mistake in such questions is unit confusion. Students may try to directly use 48 along with 18 and 8 without converting meters to centimeters, causing a wrong proportion. Another frequent error is to invert the ratio, using shadow over height in one case and height over shadow in the other case, which breaks the similarity relationship. Keeping units consistent and writing down the proportional equation carefully prevents these mistakes. Final Answer: The height of the pole is 108 m.

Question 20

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30º and 45º respectively. If the lighthouse is 100 m high, the distance between the two ships is:

Accepted Answer

273 m

Height and Distance Questions