Question 1

A balloon starts from a point P on the ground and rises vertically at a uniform speed. After 6 minutes, an observer standing at a point on level ground 450√3 metres horizontally from P finds that the angle of elevation of the balloon is 60 degrees. …

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Introduction / Context: This is a classic height and distance problem involving a balloon rising vertically at a constant speed. We are given the horizontal distance of the observer from the point where the balloon started, the time taken to reach a certain height, and the angle of elevation at that time. Using trigonometry and the definition of tangent of an angle, we can find the height of the balloon and then compute its vertical speed in metres per second. Given Data / Assumptions: The balloon starts from point P on the ground and rises vertically. The observer is on the same horizontal level as P and at a distance of 450√3 metres from P. After 6 minutes, the angle of elevation of the balloon from the observer is 60 degrees. The balloon rises at a uniform (constant) speed. We ignore effects like wind drift and assume the path is perfectly vertical. Concept / Approach: When an object is observed at an angle of elevation from a point on the ground, the tangent of that angle equals the vertical height divided by the horizontal distance between the object and the observer. Let the height of the balloon after 6 minutes be h metres. Using tan 60 degrees, we determine h in terms of the known horizontal distance 450√3. Once we have h, we use speed = distance divided by time, with time converted to seconds, to obtain the vertical speed in metres per second. Step-by-Step Solution: Step 1: Let h be the height of the balloon after 6 minutes. From the observer, horizontal distance to point P remains 450√3 metres. Step 2: The angle of elevation is 60°, so tan 60° = h / (450√3). Since tan 60° = √3, we have √3 = h / (450√3). Step 3: Multiply both sides by 450√3 to get h = 450√3 * √3 = 450 * 3 = 1350 metres. Step 4: The balloon reaches 1350 metres in 6 minutes, that is 6 × 60 = 360 seconds. Step 5: Speed = distance / time = 1350 / 360 = 3.75 metres per second. Verification / Alternative check: Approximate √3 as 1.732. Then horizontal distance is about 779.4 metres. With height 1350 metres, tan of the angle is approximately 1350 / 779.4 which is very close to 1.732, confirming that the angle is 60 degrees. Dividing 1350 by 360 gives 3.75 exactly, so the numerical computation is consistent and accurate. Why Other Options Are Wrong: Options 4.25, 4.5, 3.45, and 4.0 metres per second correspond to different heights after 6 minutes, none of which gives tan 60° equal to the ratio of height to 450√3. If we back calculate using these speeds, we do not recover a consistent angle of 60 degrees with the given horizontal distance, so they are incorrect. Common Pitfalls: A common error is to misinterpret the 450√3 metres as the height instead of the horizontal distance. Another mistake is to forget to convert 6 minutes into seconds when computing metres per second. Some learners also confuse sine and tangent, but in this type of problem, tangent is the correct ratio because it connects opposite side and adjacent side of the right angled triangle. Final Answer: The speed of the balloon is 3.75 metres per second.

Question 2

A person initially observes the top of a building at an angle of elevation of 45 degrees. After walking 100 metres straight towards the building along level ground, the angle of elevation becomes 60 degrees. What is the height of the building in met…

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Introduction / Context: This problem tests understanding of how angles of elevation change as an observer moves closer to a vertical object. The observer walks towards a building and the angle of elevation of the building top increases from 45 degrees to 60 degrees. Using trigonometry, and particularly the tangent function in two right angled triangles that share the same height, we can determine the unknown height of the building. Given Data / Assumptions: Initial angle of elevation of the top of the building is 45 degrees. After walking 100 metres towards the building, the angle of elevation becomes 60 degrees. The observer walks along level ground in a straight line directly towards the building. The building is vertical and stands on the same horizontal level as the observer. Let the height of the building be h metres. Concept / Approach: For a right angled triangle, tan of the angle of elevation equals the ratio of the height of the object to the horizontal distance between the observer and the object. If the initial horizontal distance is x metres, then tan 45 degrees gives h in terms of x, and tan 60 degrees gives h in terms of x − 100, because the observer has walked 100 metres closer. Solving this system of equations allows us to find x and then h in exact form. Step-by-Step Solution: Step 1: Let the initial distance from the building be x metres. With angle 45°, tan 45° = h / x, so 1 = h / x, which implies h = x. Step 2: After walking 100 metres towards the building, the new distance is x − 100 metres. Step 3: With angle 60°, tan 60° = h / (x − 100). Since tan 60° = √3 and h = x, we get √3 = x / (x − 100). Step 4: Cross multiply to obtain √3(x − 100) = x, which simplifies to √3 x − 100√3 = x. Step 5: Rearrange to get (√3 − 1)x = 100√3, so x = 100√3 / (√3 − 1). Step 6: Rationalise the denominator: x = 100√3(√3 + 1) / (3 − 1) = 50(3 + √3). Step 7: Since h = x, the height of the building is h = 50(3 + √3) metres. Verification / Alternative check: Approximate √3 as 1.732. Then 3 + √3 is about 4.732 and h ≈ 50 × 4.732 ≈ 236.6 metres. Initially, distance x is also 236.6 metres, giving tan 45° = 236.6 / 236.6 = 1, correct. After moving 100 metres, the new distance is about 136.6 metres, and tan 60° ≈ 236.6 / 136.6 ≈ 1.732, which matches √3 and confirms the correctness of the answer. Why Other Options Are Wrong: Option 100(√3 + 1) corresponds to a different height that does not satisfy both angle conditions. Option 150 and option 100√3 produce approximate distances that fail to give tan 45° and tan 60° simultaneously. Option 200 is a simple approximate value and does not satisfy the trigonometric equations derived from the two angles. Common Pitfalls: Learners sometimes forget that the second horizontal distance is x minus 100, not x plus 100. Another common error is to use sine instead of tangent, or to incorrectly rationalise the denominator when simplifying x. Some students also approximate √3 too early, which can lead to rounding errors and incorrect identification of the exact form of the height. Final Answer: The height of the building is 50(3 + √3) metres.

Question 3

The upper part of a vertical tree is broken by the wind and the broken top still attached to the trunk touches the ground, making an angle of 60 degrees with the ground. The horizontal distance between the root of the tree and the point where the to…

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Introduction / Context: This is a standard application of trigonometry to a practical situation. A tree has broken such that its top still remains attached and touches the ground some distance from the base. The broken part makes a known angle with the ground. We must determine the original height of the tree by modelling the situation as a right angled triangle and using the given angle and distance along the ground. Given Data / Assumptions: The tree originally stood vertically on level ground. The top portion was broken but stayed attached and now touches the ground. The broken top makes an angle of 60 degrees with the horizontal ground. The horizontal distance between the base (root) of the tree and the point where the top touches the ground is 25 metres. The tree broke at some point above the ground and not at the base. Concept / Approach: Let the original height of the tree be H metres and the height at which it broke be h metres. Then the length of the broken top is H − h. When the top touches the ground, the broken part and the ground form a right angled triangle where the broken part is a slant side and the base of the tree is another vertex. The distance between the root and the point of contact on the ground is the horizontal base of a right triangle with included angle 60 degrees between the broken top and the ground. Using trigonometry, we can express the height of the break in terms of the distance and angle, and then add the length of the top to obtain the original height. Step-by-Step Solution: Step 1: Consider triangle formed by the root O, the break point B, and the ground contact point C. The right angle is at O, and the angle at C between the broken top BC and the ground is 60°. Step 2: Let OC, the horizontal distance, be 25 metres. In triangle OBC, angle at C is 60° and angle at O is 90°. Therefore angle at B is 30°. Step 3: Using trigonometry at point C, the vertical height OB is opposite to angle 60° in triangle OBC, and BC is the broken part. Step 4: Using tan 60° = OB / OC, we can write √3 = OB / 25, so OB = 25√3 metres. Step 5: The broken part BC is the hypotenuse. Use sin 60° = OB / BC. Thus √3 / 2 = (25√3) / BC, giving BC = 50 metres. Step 6: The original height H is OB + BC = 25√3 + 50. Using √3 ≈ 1.732, H ≈ 25 × 1.732 + 50 ≈ 43.3 + 50 = 93.3 metres. Verification / Alternative check: We can verify the geometry by computing OC from the triangle. Using cos 60° = OC / BC gives 1 / 2 = 25 / BC if OC is 25 metres. Then BC = 50 metres, consistent with the earlier calculation. With OB = 25√3 and BC = 50, Pythagoras also holds, because BC² = OB² + OC² becomes 50² = (25√3)² + 25², that is 2500 = 1875 + 625, confirming the right angled triangle. Why Other Options Are Wrong: Options 84.14, 98.25, and 120.24 represent different supposed heights that do not satisfy the correct trigonometric relationships when we reconstruct the triangle. Option 80 is a simple approximate value but fails when checked with tan 60° and sin 60° for the specific 25 metre base. Only 93.3 metres is consistent with all conditions of the problem. Common Pitfalls: Some learners mistakenly consider the 25 metres as the length of the broken part instead of the horizontal distance. Others forget that the original height is the sum of the unbroken upright part and the broken top. Confusing sine and cosine for the angle at the ground is another common issue. Careful drawing of the diagram and labelling of sides helps in avoiding these mistakes. Final Answer: The original height of the tree was approximately 93.3 metres.

Question 4

A vertical tower has a height of 300 metres. When its top is viewed from the top of another smaller vertical tower, the angle of depression is 60 degrees. The horizontal distance between the bases of the two towers is 120 metres. What is the height …

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Introduction / Context: Here we have two vertical towers standing on the same horizontal ground. The taller tower has known height, and from its top an observer sees the top of the smaller tower at a given angle of depression. The horizontal distance between the two towers is also given. Using the connection between angle of depression and angle of elevation, and basic trigonometry, we can calculate the height of the smaller tower. Given Data / Assumptions: The height of the taller tower is 300 metres. The angle of depression from the top of the taller tower to the top of the smaller tower is 60 degrees. The horizontal distance between the bases of the two towers is 120 metres. Both towers stand vertically on the same level ground. We ignore effects such as earth curvature and treat the situation as a right angled triangle between tower tops and bases. Concept / Approach: An angle of depression from a horizontal line of sight at the top of one tower equals the angle of elevation from the corresponding point on the ground or from the other top. Thus, the angle of elevation of the top of the taller tower from the top of the smaller tower is also 60 degrees. Let the height of the smaller tower be h metres. The vertical difference in height between the two tower tops is 300 − h, and the horizontal distance between them is 120 metres. Using tan 60 degrees equals vertical difference divided by horizontal distance, we can solve for h. Step-by-Step Solution: Step 1: Let h be the height of the smaller tower in metres. Step 2: The difference in vertical height between the two tops is 300 − h metres. Step 3: The horizontal distance between the towers is given as 120 metres. Step 4: From the top of the taller tower, angle of depression to the top of the smaller tower is 60°. Thus, tan 60° = (300 − h) / 120. Step 5: Since tan 60° = √3, we have √3 = (300 − h) / 120, so 300 − h = 120√3. Step 6: Rearrange to find h = 300 − 120√3. Using √3 ≈ 1.732, h ≈ 300 − 120 × 1.732 ≈ 300 − 207.84 = 92.16 metres, which is approximately 92.15 metres. Verification / Alternative check: Check back in the tangent relation. Vertical difference 300 − 92.16 ≈ 207.84 metres. Then tan 60° should be 207.84 / 120 ≈ 1.732, which matches √3. The geometry is consistent with a right triangle having base 120 and opposite side approximately 207.84 for a 60 degree angle, confirming the solution. Why Other Options Are Wrong: Options 88.24, 106.71, and 112.64 correspond to different vertical differences that do not lead to tan 60° when divided by 120. Option 100 would give a difference of 200 metres, making tan 60° approximately 1.666, which is not equal to √3. Only about 92.15 metres satisfies all conditions of the problem. Common Pitfalls: One common mistake is to misinterpret angle of depression and to use 300 + h as the vertical distance instead of 300 − h. Another error is to confuse distance between bases with distance between tops. Some learners also make algebraic mistakes when isolating h, or approximate √3 too roughly, which can mislead them towards an incorrect option when numerical values are close. Final Answer: The height of the smaller tower is approximately 92.15 metres.

Question 5

From a point P, the angle of elevation of a vertical tower is such that its tangent is 3/4. After walking 560 metres straight towards the tower along level ground, the tangent of the angle of elevation becomes 4/3. What is the height of the tower in…

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Introduction / Context: This problem examines how the angle of elevation of a tower changes as an observer moves closer to it. Instead of giving the actual angles, the question provides the values of their tangents, which is equivalent information. Using the tangent ratio definition, we can set up equations in terms of the tower height and horizontal distances, and then solve them to find the height of the tower. Given Data / Assumptions: From point P, tan of the angle of elevation of the tower is 3/4. After walking 560 metres directly towards the tower, tan of the new angle of elevation is 4/3. The tower is vertical and stands on level ground. The observer walks along a straight horizontal line towards the tower. We ignore any height of the observer and assume measurements from ground level. Concept / Approach: If the height of the tower is h metres and the initial horizontal distance from point P is x metres, then tan θ1 = h / x. Given tan θ1 = 3/4, we obtain a relation between h and x. After walking 560 metres closer, the distance becomes x − 560 metres, and tan θ2 = h / (x − 560) is given as 4/3. These two equations in terms of h and x can be solved simultaneously to find x and then h. Step-by-Step Solution: Step 1: Let h be the height of the tower and x be the initial horizontal distance from point P. Step 2: From the first position, tan θ1 = h / x = 3 / 4, so h = (3/4)x. Step 3: From the second position, the distance to the tower is x − 560. Here tan θ2 = h / (x − 560) = 4 / 3. Step 4: Substitute h = (3/4)x into the second relation: (3/4)x / (x − 560) = 4 / 3. Step 5: Simplify: (3x) / (4(x − 560)) = 4 / 3. Cross multiply to get 9x = 16(x − 560). Step 6: Expand and solve: 9x = 16x − 8960, so 7x = 8960 and x = 1280 metres. Step 7: Substitute back to find h: h = (3/4)x = (3/4) × 1280 = 960 metres. Verification / Alternative check: Verify both tangents. Initial distance 1280 metres with height 960 metres gives tan θ1 = 960 / 1280 = 3 / 4, correct. After walking 560 metres, the distance becomes 1280 − 560 = 720 metres, so tan θ2 = 960 / 720 = 4 / 3, which matches the given value. This confirms that h = 960 metres is consistent with all data. Why Other Options Are Wrong: If height were 720 metres, the first tangent would be 720 / 1280 which is not 3 / 4. A height of 840 metres or 1030 metres fails to produce the correct ratio 4 / 3 for the second position when the displacement is 560 metres. Option 880 is another trial value that does not satisfy both tangent conditions together. Only 960 metres satisfies both simultaneously. Common Pitfalls: Students sometimes forget that the new distance from the tower is x minus 560 and not x plus 560. Another frequent error is in cross multiplication or simplifying the resulting equation. It is also easy to misinterpret tangents and accidentally invert the ratios, writing x / h instead of h / x. Careful algebra and clear understanding of the tangent definition help avoid these mistakes. Final Answer: The height of the tower is 960 metres.

Question 6

The tops of two vertical poles of heights 60 metres and 35 metres are connected by a straight rope. The rope makes an angle with the horizontal such that tan of this angle is 5/9. What is the horizontal distance between the two poles in metres?

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Introduction / Context: This question involves two vertical poles of different heights connected at their tops by a rope that is inclined to the horizontal. The tangent of the angle that the rope makes with the horizontal is given. The difference in heights of the poles and the tangent value allow us to compute the horizontal distance between the poles using basic trigonometry and the definition of tangent. Given Data / Assumptions: One pole has a height of 60 metres. The other pole has a height of 35 metres. The tops of the poles are joined by a straight rope. The rope makes an angle θ with the horizontal such that tan θ = 5 / 9. The poles stand vertically on the same level ground. Concept / Approach: Let the horizontal distance between the two poles be d metres. The vertical difference between their tops is equal to the difference in their heights. The rope is the slant side of a right angled triangle whose opposite side is this vertical difference and whose adjacent side is the horizontal distance d. Since tan θ is defined as opposite side divided by adjacent side, we can write tan θ = (vertical difference) / d. Substituting the known values, we can solve for d. Step-by-Step Solution: Step 1: Calculate the vertical difference in height between the two poles: 60 − 35 = 25 metres. Step 2: Let d be the horizontal distance between the two poles in metres. Step 3: From the given information, tan θ = 5 / 9 and tan θ also equals vertical difference divided by horizontal distance, so 5 / 9 = 25 / d. Step 4: Cross multiply to obtain 5d = 9 × 25 = 225. Step 5: Solve for d: d = 225 / 5 = 45 metres. Verification / Alternative check: Check the tangent value using the found distance. With opposite side 25 metres and adjacent side 45 metres, tan θ = 25 / 45 = 5 / 9, which exactly matches the given condition. This confirms that the horizontal distance has been computed correctly. The geometry matches a right angled triangle formed by the difference in heights and the distance between the poles. Why Other Options Are Wrong: If the distance were 63 metres or 30 metres, the tangent of the angle formed by the rope would be 25 / 63 or 25 / 30, neither of which equals 5 / 9. A distance of 25 metres would lead to tan θ = 1, and a distance of 40 metres would give tan θ = 25 / 40 = 5 / 8. None of these satisfy the given condition tan θ = 5 / 9. Common Pitfalls: A common error is to use the sum of the heights instead of the difference, but the rope connects the tops of the poles, so only their height difference forms the opposite side. Another mistake is to invert the ratio and use d / 25 rather than 25 / d for tan θ. Students may also confuse which segment is horizontal and which is vertical in the right angled triangle. Drawing a clear diagram often prevents these misunderstandings. Final Answer: The horizontal distance between the two poles is 45 metres.

Height and Distance Questions