Question 1

The top of a tree is broken by a storm and the broken part touches the ground at a point 15 m from the base of the tree. If the tree is broken at a height of 8 m above the ground and the broken top remains attached, what was the original total heigh…

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Introduction: This is a classic broken-tree (or broken-pole) trigonometry-style question, but it can be solved using only the Pythagoras theorem. The intact lower part of the tree remains vertical, while the broken top forms a right-angled triangle with the ground and the remaining vertical part. Given Data / Assumptions: The tree is broken at a height of 8 m above the ground. The top of the broken part touches the ground at a point 15 m horizontally from the base. The broken top remains attached to the stump, forming a straight segment from the break point to the ground. The tree was vertical before breaking and the ground is level. Concept / Approach: We treat the situation as a right-angled triangle. The remaining vertical part of the tree is one leg of the triangle, the horizontal distance from the base to where the top touches the ground is the other leg, and the broken top itself forms the hypotenuse. Once the length of the broken part is found, we add it to the unbroken stump height to get the original height. Step-by-Step Solution: Step 1: Let the length of the broken top be L.Step 2: The vertical leg is 8 m (height of the break point).Step 3: The horizontal leg is 15 m (distance from base to where the top touches the ground).Step 4: Apply Pythagoras theorem: L² = 8² + 15².Step 5: Compute: 8² = 64 and 15² = 225, so L² = 64 + 225 = 289.Step 6: Take the square root: L = √289 = 17 m.Step 7: Original height of the tree = stump height + broken part length = 8 + 17 = 25 m. Verification / Alternative check: Check the triangle: sides 8, 15, and 17 form a well-known Pythagorean triple (8² + 15² = 17²). This confirms our triangle is right-angled and that 17 m is the correct length of the broken part, giving a total height of 25 m. Why Other Options Are Wrong: Heights like 17 m or 23 m account only for part of the tree, not both sections. Values like 27 m or 30 m are obtained if the Pythagoras theorem is misapplied or if 8 and 15 are incorrectly combined (for example, just adding them without considering the hypotenuse). They do not satisfy the right-triangle relationship. Common Pitfalls: A frequent mistake is to think that the original height is simply 15 + 8 = 23 m or to treat 15 m as the hypotenuse. Always identify which segment is slant (hypotenuse) and which are horizontal and vertical legs before applying Pythagoras theorem. Final Answer: The original height of the tree was 25 m.

Question 2

A ladder 13 m long just reaches a window that is 12 m above the ground on one side of a street. Keeping its foot fixed at the same point on the ground, the ladder is then turned to the other side of the street and now just reaches a window 5 m high.…

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Introduction: This problem uses the Pythagoras theorem in two different right-angled triangles formed by a ladder leaning against buildings on opposite sides of a street. Because the ladder length remains fixed, and only the vertical heights change, we can compute the horizontal distances on each side and add them to get the street's width. Given Data / Assumptions: Length of the ladder (hypotenuse) = 13 m. Height of the window on one side = 12 m. Height of the window on the opposite side = 5 m. The ladder foot stays at the same point on the ground between the two buildings. The ground is level and the buildings are vertical, forming right-angled triangles with the ladder. Concept / Approach: On each side, the ladder, the building height, and the ground form a right-angled triangle. We can find the horizontal distance from the ladder foot to each building using: (horizontal distance)² + (height)² = (ladder length)².The width of the street is the sum of the horizontal distances to the two buildings. Step-by-Step Solution: Step 1: First side (12 m window): Let x₁ be the horizontal distance.Step 2: Apply Pythagoras: x₁² + 12² = 13² ⇒ x₁² + 144 = 169 ⇒ x₁² = 25 ⇒ x₁ = 5 m.Step 3: Second side (5 m window): Let x₂ be the horizontal distance.Step 4: Apply Pythagoras: x₂² + 5² = 13² ⇒ x₂² + 25 = 169 ⇒ x₂² = 144 ⇒ x₂ = 12 m.Step 5: Width of the street = x₁ + x₂ = 5 + 12 = 17 m. Verification / Alternative check: You can verify that the pairs (5,12,13) and (12,5,13) each satisfy the Pythagoras theorem: 5² + 12² = 25 + 144 = 169 = 13². This confirms that the horizontal distances are calculated correctly and that adding them gives the true street width. Why Other Options Are Wrong: Options like 14 m, 15 m, 16 m, or 19 m arise if you miscompute one of the horizontal distances or incorrectly subtract instead of adding them. None of these satisfy the two right triangles with the same 13 m ladder and the given heights of 12 m and 5 m. Common Pitfalls: Students sometimes think the street width is the difference, not the sum, of the two horizontal distances, or they swap the roles of heights and hypotenuse in the Pythagoras formula. Always remember that the ladder is the hypotenuse and that the buildings lie on opposite sides of the fixed foot of the ladder, so distances must be added. Final Answer: The width of the street is 17 m.

Question 3

From a point P on level ground, the angle of elevation of the top of a tower is such that tan θ = 3/4. After walking 560 m straight towards the tower, the angle of elevation changes so that tan θ = 4/3. What is the height (in metres) of the tower?

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Introduction: This is a height and distance problem where the observer changes position and the tangent of the angle of elevation is given instead of the angle itself. We use right-angled triangles and the definition of tangent to set up equations involving the height of the tower and the horizontal distances. Given Data / Assumptions: From point P, tan θ = 3/4 for the angle of elevation. The observer walks 560 m towards the tower to a new point. From the new point, tan θ = 4/3 for the angle of elevation. The ground is level and the tower is vertical. We must find the height h of the tower. Concept / Approach: Let the initial horizontal distance from P to the foot of the tower be d. Then, using tan θ = opposite / adjacent: From P: h / d = 3/4.From the nearer point: h / (d − 560) = 4/3.These give two equations in h and d. Solving this system yields the height of the tower. Step-by-Step Solution: Step 1: From P: h = (3/4) * d.Step 2: From second point: h = (4/3) * (d − 560).Step 3: Equate the two expressions for h:(3/4) * d = (4/3) * (d − 560).Step 4: Multiply both sides by 12 to clear denominators: 9d = 16(d − 560).Step 5: Expand right side: 9d = 16d − 8960.Step 6: Rearrange: 9d − 16d = −8960 ⇒ −7d = −8960 ⇒ d = 1280 m.Step 7: Substitute into h = (3/4)d: h = (3/4) * 1280 = 3 * 320 = 960 m. Verification / Alternative check: Check both positions: From P, tan θ = h/d = 960/1280 = 3/4, correct. From the nearer point, distance = 1280 − 560 = 720 m, so tan θ = 960/720 = 4/3, also correct. This verifies that h = 960 m is consistent with both given tangents. Why Other Options Are Wrong: Heights like 720 m or 840 m are obtained if you mistakenly use 560 instead of d − 560 or confuse the tangent ratios. 1030 m or 640 m do not satisfy both tangent conditions when checked, so they cannot be the true height. Common Pitfalls: Typical errors include misplacing the 560 m term (adding instead of subtracting) or inverting the tangent ratios (using 4/3 in the first case and 3/4 in the second). Careful equation setup and solving are essential. Final Answer: The height of the tower is 960 m.

Question 4

On level ground there is a vertical tower with a flagpole mounted on its top. From a point 9 m away from the foot of the tower along the ground, the angle of elevation of the top of the tower (bottom of the flagpole) is 30°, and the angle of elevati…

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Introduction: This problem involves two angles of elevation observed from the same point to two different heights on the same vertical line: the top of a tower and the top of a flagpole mounted on that tower. By using basic trigonometry, we can separately determine the height of the tower and the extra height due to the flagpole, then subtract to find the flagpole's height. Given Data / Assumptions: Distance from observation point to foot of the tower = 9 m. Angle of elevation to the top of the tower (bottom of flagpole) = 30°. Angle of elevation to the top of the flagpole = 60°. The tower and flagpole together form a straight vertical line. The ground is horizontal and the tower is vertical. Concept / Approach: Let H be the height of the tower and (H + h) be the total height including the flagpole, where h is the height of the flagpole. Using tangent for right-angled triangles from the observation point: tan 30° = H / 9.tan 60° = (H + h) / 9.We use tan 30° = 1/√3 and tan 60° = √3 to solve for H and h. Step-by-Step Solution: Step 1: From tan 30° = H/9: 1/√3 = H / 9 ⇒ H = 9 / √3.Step 2: Rationalise: H = 9 / √3 = 9√3 / 3 = 3√3 m.Step 3: From tan 60° = (H + h)/9: √3 = (H + h)/9 ⇒ H + h = 9√3.Step 4: Substitute H = 3√3: 3√3 + h = 9√3.Step 5: Solve for h: h = 9√3 − 3√3 = 6√3 m. Verification / Alternative check: If the flagpole height is 6√3 m and the tower is 3√3 m, the total height is 9√3 m. Checking the tangents: for 30°, H/9 = (3√3)/9 = √3/3 = 1/√3, correct; for 60°, (H + h)/9 = (9√3)/9 = √3, also correct. Why Other Options Are Wrong: Options such as 5√3 m or 4√3 m do not lead to the correct total height and so fail one or both tangent conditions. 6√2 m and 6√5 m are not related to standard √3-based tangent values for 30° and 60° and are therefore inconsistent with the trigonometric setup. Common Pitfalls: Learners sometimes forget that the 60° angle corresponds to the combined height of tower plus flagpole, not just the flagpole. Others mix up tan 30° and tan 60°, or mistakenly use sine instead of tangent. Always clearly define the heights for each angle before writing the equations. Final Answer: The height of the flagpole is 6√3 m.

Question 5

Two ships are on opposite sides of a vertical lighthouse. The angles of elevation of the top of the lighthouse from the two ships are 30° and 45°, respectively. If the lighthouse is 100 m high (take √3 ≈ 1.73), what is the distance (in metres) betwe…

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Introduction: This is a straightforward height and distance problem involving a lighthouse and two ships on opposite sides of it. By using basic trigonometry, we can find the horizontal distance from each ship to the lighthouse and then add these distances to get the total separation between the ships. Given Data / Assumptions: Height of the lighthouse = 100 m. Angle of elevation from ship 1 = 30°. Angle of elevation from ship 2 = 45°. The ships are on opposite sides of the lighthouse. The sea surface is taken to be level. Approximate √3 as 1.73 if needed for numerical values. Concept / Approach: Let d₁ be the distance from ship 1 to the lighthouse foot and d₂ be the distance from ship 2. For each right-angled triangle, we use: tan θ = height / distance.We then compute d₁ and d₂ and add them to obtain the separation between the two ships. Step-by-Step Solution: Step 1: For ship 1 (30°): tan 30° = 1/√3 = 100 / d₁ ⇒ d₁ = 100√3.Step 2: For ship 2 (45°): tan 45° = 1 = 100 / d₂ ⇒ d₂ = 100 m.Step 3: Distance between the ships = d₁ + d₂ = 100√3 + 100.Step 4: Using √3 ≈ 1.73, compute 100√3 ≈ 173.Step 5: Therefore, separation ≈ 173 + 100 = 273 m. Verification / Alternative check: Check with rough sketch: the ship with angle 45° must be closer, so 100 m makes sense. The other ship at distance about 173 m also fits a shallower 30° angle. Summing the two distances gives a realistic total separation. Why Other Options Are Wrong: 173 m represents only one leg (distance of the farther ship) and ignores the other ship. 200 m or 300 m do not match the sum of 100 and 100√3. 250 m is simply a distractor and does not correspond to any correct trigonometric combination from the given data. Common Pitfalls: Common mistakes include subtracting the distances instead of adding them, or assigning the wrong angle to the wrong distance (for example, using 45° for the farther ship). Remember: larger angle → closer distance; and since the ships are on opposite sides, the total separation is the sum of the two distances. Final Answer: The distance between the two ships is approximately 273 m.

Question 6

A man standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is 60°. When he walks 36 m directly away from the river along a straight line perpendicular to the bank, the angle of elevation of t…

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Introduction: This is a classic river-width problem using trigonometry. The observer changes position along a line perpendicular to the bank and sees the angle of elevation of the top of a tree change. From this, we can set up two tangent relations and solve for the distance from the tree to the bank, which is the river's breadth. Given Data / Assumptions: The man initially stands at point A on the bank. Angle of elevation of the tree top from A = 60°. He then walks 36 m back to point B, directly away from the river. Angle of elevation from B = 30°. The tree stands vertically on the opposite bank, and the ground is level. Concept / Approach: Let x be the breadth of the river (distance from A to the foot of the tree) and h be the height of the tree. From trigonometry in right-angled triangles, we have: tan 60° = h / x.tan 30° = h / (x + 36).Substituting tan 60° = √3 and tan 30° = 1/√3, we can solve these two equations to find x. Step-by-Step Solution: Step 1: From the first position: tan 60° = h/x ⇒ √3 = h/x ⇒ h = √3 * x.Step 2: From the second position: tan 30° = h/(x + 36) ⇒ 1/√3 = h/(x + 36) ⇒ h = (x + 36)/√3.Step 3: Equate the two expressions for h: √3 x = (x + 36)/√3.Step 4: Multiply both sides by √3 to clear the denominator: 3x = x + 36.Step 5: Rearrange: 3x − x = 36 ⇒ 2x = 36 ⇒ x = 18 m. Verification / Alternative check: Using x = 18, compute h from h = √3 x: h = 18√3. From the farther point, distance is x + 36 = 54 m. Check tan 30°: h/(x + 36) = (18√3)/54 = √3/3 = 1/√3, which matches tan 30°. Both angles are consistent with x = 18 m as the river width. Why Other Options Are Wrong: Values like 15 m, 16 m, 11 m, or 20 m fail to satisfy both trigonometric conditions simultaneously. When used in the equations, they produce conflicting tree heights h, indicating they are not correct solutions. Common Pitfalls: Students sometimes treat 36 m as being added on the tree side instead of the observer side or misapply tangent values (interchanging √3 with 1/√3). It is also easy to forget that the second distance is x + 36, not x − 36. Drawing a clear diagram helps avoid these mistakes. Final Answer: The breadth of the river is 18 m.

Question 7

A kite is flying in the sky, attached to a straight string that is pulled taut. The length of the string from a point on the ground to the kite is 420 m, and the angle of elevation of the string with the horizontal ground is 30°. Assuming there is n…

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Introduction: This problem is a direct application of basic trigonometry to a height and distance context involving a flying kite. The string forms the hypotenuse of a right-angled triangle, and the angle between the string and the horizontal ground is given. We must find the vertical height of the kite using the sine function. Given Data / Assumptions: Length of the string (hypotenuse) = 420 m. Angle of elevation between string and ground = 30°. The string is taut (no slack), so it forms a straight line. The ground is level and the kite is directly above some point on the ground. Concept / Approach: In a right-angled triangle, the sine of an angle is defined as opposite side divided by hypotenuse. Here, the opposite side is the vertical height h of the kite, and the hypotenuse is the string length 420 m. Thus: sin 30° = h / 420.We use the known value sin 30° = 1/2 to solve for h. Step-by-Step Solution: Step 1: Write the sine relation: sin 30° = h / 420.Step 2: Substitute sin 30° = 1/2, giving 1/2 = h / 420.Step 3: Solve for h: h = 420 * (1/2) = 210 m. Verification / Alternative check: We can also find the horizontal distance to the point directly under the kite: using cos 30° = adjacent / hypotenuse, we get adjacent = 420 * cos 30° = 420 * (√3/2) = 210√3 m. This is consistent with a standard 30°–60°–90° triangle, where the sides are in the ratio 1 : √3 : 2. Thus, our value h = 210 m fits the triangle perfectly. Why Other Options Are Wrong: 140√3 and 210√3 represent different legs of a 30°–60°–90° triangle but not the vertical height for a 30° angle with hypotenuse 420. Heights like 150 m or 120 m do not satisfy sin 30° = h/420 when checked: 150/420 and 120/420 are not equal to 1/2. Common Pitfalls: A common mistake is to mistakenly use cosine instead of sine for the vertical height, or to confuse which side (opposite or adjacent) corresponds to the given angle. Remember: with the angle at the ground and the vertical height opposite that angle, sine is the correct ratio to use. Final Answer: The height of the kite above the ground is 210 m.

Question 8

From the top of a 60 m high vertical tower, the angles of depression of the top and bottom of a vertical pole standing on the same level ground are 45° and 60°, respectively. What is the height (in metres) of the pole?

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Introduction: This question involves angles of depression from the top of a tower to both the top and bottom of a pole. The geometry forms two right-angled triangles sharing the same horizontal distance between the tower and the pole. Using tangent relations for the angles of depression, we can determine both the horizontal distance and the height of the pole. Given Data / Assumptions: Height of the tower = 60 m. Angle of depression to the top of the pole = 45°. Angle of depression to the bottom (foot) of the pole = 60°. The tower and the pole stand on the same horizontal ground. Both are vertical, and the horizontal distance between them is the same for both lines of sight. Concept / Approach: Angles of depression from the top of the tower equal the corresponding angles of elevation from the ground. Let the horizontal distance between the tower and the pole be d, and the height of the pole be h. Then: From the top to the foot: tan 60° = (60 − 0) / d = 60 / d.From the top to the top of the pole: tan 45° = (60 − h) / d.Using tan 60° = √3 and tan 45° = 1, we can solve for d and h. Step-by-Step Solution: Step 1: From the foot of the pole: tan 60° = 60 / d ⇒ √3 = 60 / d ⇒ d = 60 / √3 = 20√3 m.Step 2: From the top of the pole: tan 45° = (60 − h) / d ⇒ 1 = (60 − h) / d ⇒ 60 − h = d.Step 3: Substitute d = 20√3 into 60 − h = 20√3.Step 4: Rearrange to find h: h = 60 − 20√3.Step 5: Factor out 20: h = 20(3 − √3) m. Verification / Alternative check: We can check numerically using √3 ≈ 1.732: h ≈ 20(3 − 1.732) = 20 * 1.268 ≈ 25.36 m. With d ≈ 20√3 ≈ 34.64 m, tan 60° ≈ 60/34.64 ≈ 1.732, correct, and tan 45° ≈ (60 − 25.36)/34.64 ≈ 34.64/34.64 ≈ 1, also correct. Why Other Options Are Wrong: 20(√3 + 1) m and 20(3√3) m are much larger than the tower height or inconsistent with the tangent equations. 60 − 20√3 m is algebraically the same as 20(3 − √3) m, but the fully factored form is clearer in this context; any expression not equivalent to this value will not satisfy both angle conditions. Common Pitfalls: Learners often mix up which vertical difference to use for which angle or forget that the vertical drop to the top of the pole is (60 − h), not h. Another common mistake is to use distances with the wrong angle (e.g., using tan 45° for the foot). Drawing both triangles clearly and labeling heights and distances carefully helps avoid confusion. Final Answer: The height of the pole is 20(3 − √3) m.

Question 9

A vertical pole is 15 m high. If the Sun’s elevation (altitude) changes from 30° to 60°, what is the difference (in metres) between the lengths of the shadows of the pole corresponding to these two angles of elevation?

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Introduction: This question tests your understanding of how the length of a shadow of a vertical object changes with the Sun’s angle of elevation. Using basic trigonometry, we express the shadow length in terms of the height and the tangent of the Sun’s altitude for each case, and then find the difference between the two shadow lengths. Given Data / Assumptions: Height of the vertical pole = 15 m. First altitude of the Sun = 30°. Second altitude of the Sun = 60°. The pole stands perfectly vertical on level ground. Concept / Approach: For a vertical object of height h and Sun’s altitude θ, the shadow length L on level ground is given by: tan θ = h / L ⇒ L = h / tan θ.We compute L₁ for θ = 30° and L₂ for θ = 60°, then find the difference L₁ − L₂. Step-by-Step Solution: Step 1: Let h = 15 m.Step 2: For θ = 30°, tan 30° = 1/√3.Step 3: Shadow length L₁ = h / tan 30° = 15 / (1/√3) = 15√3.Step 4: For θ = 60°, tan 60° = √3.Step 5: Shadow length L₂ = h / tan 60° = 15 / √3 = (15√3)/3 = 5√3.Step 6: Difference in shadow lengths: L₁ − L₂ = 15√3 − 5√3 = 10√3 m. Verification / Alternative check: Using √3 ≈ 1.732, approximate L₁ ≈ 15 * 1.732 ≈ 25.98 m, L₂ ≈ 5 * 1.732 ≈ 8.66 m. The difference is about 17.32 m, which equals 10 * 1.732 ≈ 10√3 m, matching our exact expression. Why Other Options Are Wrong: 7.5 m and 15 m ignore the trigonometric dependence on the angles and are far smaller than the true difference. 5√3 m is only the shorter shadow length itself, not the difference. 20 m does not correspond to any correct trigonometric combination derived from the given angles and pole height. Common Pitfalls: Students sometimes incorrectly subtract the heights instead of the shadow lengths or confuse tan θ with its reciprocal. Remember: for a fixed height, as the angle of elevation increases, the shadow becomes shorter. Correctly using L = h / tan θ for each angle avoids such errors. Final Answer: The difference between the lengths of the shadows is 10√3 m.

Question 10

From a point on level ground that is 129 m away from the foot of a vertical cliff, the angle of elevation of the top of the cliff is observed to be 30°. What is the height (in metres) of the cliff?

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Introduction: This is a straightforward height and distance question. A cliff is observed from a known horizontal distance, and the angle of elevation of its top is given. Using basic trigonometry, we can compute the height of the cliff from these data. Given Data / Assumptions: Horizontal distance from observation point to foot of cliff = 129 m. Angle of elevation to the top of the cliff = 30°. The cliff is vertical and the ground is level. Concept / Approach: In a right-angled triangle formed by the cliff height, the ground distance, and the line of sight, the tangent of the angle of elevation is: tan θ = (height of cliff) / (horizontal distance).Here, θ = 30°, horizontal distance = 129 m. We use tan 30° = 1/√3 to find the height h. Step-by-Step Solution: Step 1: Let h be the cliff height.Step 2: Use tan 30° = h / 129.Step 3: Substitute tan 30° = 1/√3, so 1/√3 = h / 129.Step 4: Solve for h: h = 129 / √3.Step 5: Rationalise the denominator: h = (129√3) / 3 = 43√3 m. Verification / Alternative check: Approximate numerically with √3 ≈ 1.732: h ≈ 43 * 1.732 ≈ 74.5 m. Check tan 30°: h / 129 ≈ 74.5 / 129 ≈ 0.577 / 1.732 ≈ 1/√3, which confirms that the angle is indeed 30° and that the calculation is consistent. Why Other Options Are Wrong: 45√3 m, 47√3 m, or 50√3 m correspond to different heights that would yield tangents larger than 1/√3 when divided by 129 m. 43 m is too small and does not fit tan 30°. Only 43√3 m satisfies the relationship h = 129 * tan 30° exactly. Common Pitfalls: Students sometimes use sine or cosine instead of tangent for height-and-distance problems, or they forget to rationalise the denominator, leaving the answer as 129/√3. While 129/√3 is correct, the simplified form 43√3 is usually preferred and also matches the options given in such exams. Final Answer: The height of the cliff is 43√3 metres.

Question 11

A vertical tower is 50 m high. Its shadow on level ground is x metres shorter when the Sun's altitude is 45° than when the Sun's altitude is 30°. What is the value of x (the difference in shadow lengths) in metres?

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Introduction: This problem explores how the length of a shadow cast by a vertical tower changes when the Sun's altitude (angle of elevation) changes. We use the tangent function to find the shadow lengths at two angles and then compute the difference between them, which is given as x. Given Data / Assumptions: The tower is vertical and 50 m high. Sun’s altitude (angle of elevation) in case 1 = 30°. Sun’s altitude in case 2 = 45°. The shadow at 45° is x metres shorter than the shadow at 30°. Ground is level and there are no obstructions. Concept / Approach: For a vertical object of height h and Sun’s altitude θ, the shadow length L satisfies: tan θ = h / L ⇒ L = h / tan θ.We calculate shadow lengths L₃₀ and L₄₅ for 30° and 45° and then compute x = L₃₀ − L₄₅. Step-by-Step Solution: Step 1: Let h = 50 m.Step 2: For θ = 30°, tan 30° = 1/√3.Step 3: Shadow length at 30°: L₃₀ = 50 / tan 30° = 50 / (1/√3) = 50√3.Step 4: For θ = 45°, tan 45° = 1.Step 5: Shadow length at 45°: L₄₅ = 50 / tan 45° = 50 / 1 = 50.Step 6: The shadow at 45° is shorter by x: x = L₃₀ − L₄₅ = 50√3 − 50.Step 7: Factor out 50: x = 50(√3 − 1) metres. Verification / Alternative check: Using √3 ≈ 1.732, we get x ≈ 50 * (1.732 − 1) = 50 * 0.732 ≈ 36.6 m. This is a reasonable reduction in shadow length when the Sun’s altitude increases from 30° to 45°. It is also consistent with the general behaviour that shadows become shorter as the Sun appears higher in the sky. Why Other Options Are Wrong: 50√3 is the longer shadow itself, not the difference. 50 is the shorter shadow length at 45°, not x. 50(√3 + 1) would correspond to a sum, not a difference. 25(√3 − 1) is exactly half of the correct value and does not come from any correct step in the derivation. Common Pitfalls: Some students mistakenly compute x as L₄₅ − L₃₀, making it negative, or mix up tan 30° and tan 45°. Others forget to factor the expression, failing to recognise 50(√3 − 1) as the simplified form. Always check which shadow is longer and subtract in the correct order. Final Answer: The value of x is 50(√3 − 1) metres.

Question 12

For a vertical tower standing on level ground, the length of its shadow is √3 times the actual height of the tower. What is the angle of elevation of the Sun at that moment?

Accepted Answer

Introduction: This conceptual trigonometry question links shadow length and object height to the Sun’s angle of elevation. When the ratio of height to shadow length is known, we can directly use the definition of tangent to find the angle. Recognising standard trigonometric ratios is very helpful here. Given Data / Assumptions: The tower is vertical and stands on horizontal ground. Let the tower height be h. Given: Shadow length L = √3 * h. Angle of elevation of the Sun is θ. Concept / Approach: In a right-angled triangle formed by the tower, its shadow, and the line of sight to the top, the tangent of the Sun’s altitude θ is: tan θ = (opposite side) / (adjacent side) = height / shadow = h / L.We substitute L = √3 * h and simplify to identify which standard angle has this tangent. Step-by-Step Solution: Step 1: Use the formula: tan θ = h / L.Step 2: Substitute L = √3 * h: tan θ = h / (√3 * h).Step 3: Cancel h (assuming h ≠ 0): tan θ = 1 / √3.Step 4: Recall standard trigonometric values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.Step 5: Therefore, θ = 30°. Verification / Alternative check: We can quickly verify: for θ = 30°, tan 30° = 1/√3, so L = h / tan 30° = h / (1/√3) = √3 * h. This matches the given condition that the shadow length is √3 times the height of the tower. Why Other Options Are Wrong: At 45°, tan 45° = 1, giving L = h, not √3 * h. At 60°, tan 60° = √3, giving L = h / √3, which is shorter than the height. The option "none" is incorrect because a standard angle (30°) clearly satisfies the condition. 75° has a tangent significantly greater than 1, also contradicting the given ratio. Common Pitfalls: A common error is to invert the ratio and treat tan θ as L/h instead of h/L, which leads to the wrong angle. Another mistake is misremembering standard tangent values. To avoid confusion, always set up tan θ = opposite / adjacent with height as opposite and shadow as adjacent, then match the ratio to known values. Final Answer: The angle of elevation of the Sun is 30°.

Question 13

Two persons stand on opposite sides of a temple that is 75 m high. The angles of elevation of the top of the temple from these two persons are 30° and 60°, respectively. Assuming the ground is level, what is the distance (in metres) between the two …

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Introduction: This height and distance problem involves two observers on opposite sides of a temple of known height, each seeing the top under different angles of elevation. By using right-angled triangles and the tangent function, we can determine how far each observer is from the temple and then sum these distances to find the distance between the two persons. Given Data / Assumptions: Height of the temple = 75 m. Angle of elevation from person A = 30°. Angle of elevation from person B (on the other side) = 60°. The ground is horizontal and the temple is vertical. The two observers are on the same straight line passing through the base of the temple but on opposite sides. Concept / Approach: Let d₁ be the distance of person A (seeing 30°) from the temple and d₂ be the distance of person B (seeing 60°). For each right triangle, we use: tan θ = height / distance.The total distance between the two persons is d₁ + d₂. Step-by-Step Solution: Step 1: For person A (30°): tan 30° = 1/√3 = 75 / d₁ ⇒ d₁ = 75√3.Step 2: For person B (60°): tan 60° = √3 = 75 / d₂ ⇒ d₂ = 75 / √3 = 25√3.Step 3: Distance between the two persons = d₁ + d₂ = 75√3 + 25√3 = 100√3.Step 4: Using √3 ≈ 1.732, compute: 100√3 ≈ 100 * 1.732 ≈ 173.2 m. Verification / Alternative check: Check that the distances are reasonable: The observer with the smaller angle (30°) must be farther away (75√3 ≈ 129.9 m), and the observer with the larger angle (60°) must be closer (25√3 ≈ 43.3 m). Adding them gives ≈ 173.2 m, which fits the geometry and the trigonometric expectations. Why Other Options Are Wrong: 100 m or 200 m do not correspond to 100√3 and do not match the correct addition of 75√3 and 25√3. 157.7 m and 273.2 m are distractors based on incorrect combinations or approximations of √3. Only 173.2 m is equal to 100√3 and satisfies all given conditions. Common Pitfalls: A typical error is to subtract the distances instead of adding them, forgetting that the observers are on opposite sides of the temple. Another mistake is to reverse the use of 30° and 60°, leading to inverted distances. Always remember: smaller angle → farther distance; larger angle → closer distance. Final Answer: The distance between the two persons is approximately 173.2 m.

Question 14

From the top of a 20 m high building, the angle of elevation of the top of a nearby vertical tower is 60°, and the angle of depression of the foot of the same tower is 45°. (Take √3 ≈ 1.732.) What is the height (in metres) of the tower?

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Introduction: This question combines an angle of elevation and an angle of depression from the top of a building to find the height of a nearby tower. By using right-angled triangles and the tangent function, we can find the horizontal distance between the building and the tower and then use it to compute the tower’s total height. Given Data / Assumptions: Height of the building = 20 m. Angle of elevation from the building top to the tower top = 60°. Angle of depression from the building top to the tower foot = 45°. The building and tower stand on the same horizontal ground. The tower is vertical and the ground is level. Concept / Approach: Let H be the height of the tower and d be the horizontal distance between the building and the tower. From the top of the building: To the tower foot: tan 45° = (vertical drop) / d = 20 / d.To the tower top: tan 60° = (H − 20) / d.We first find d from the 45° relation and then substitute into the 60° relation to find H. Step-by-Step Solution: Step 1: From angle of depression 45° to the foot: tan 45° = 1 = 20 / d ⇒ d = 20 m.Step 2: From angle of elevation 60° to the top: tan 60° = √3 = (H − 20) / d.Step 3: Substitute d = 20: √3 = (H − 20) / 20.Step 4: Rearrange to solve for H: H − 20 = 20√3 ⇒ H = 20√3 + 20.Step 5: Using √3 ≈ 1.732, compute H ≈ 20 * 1.732 + 20 = 34.64 + 20 = 54.64 m. Verification / Alternative check: Check back with d = 20 m: For the top, (H − 20)/d ≈ (54.64 − 20)/20 = 34.64/20 ≈ 1.732, consistent with tan 60°. For the foot, 20/d = 20/20 = 1, consistent with tan 45°. Both angles are satisfied accurately. Why Other Options Are Wrong: Values like 45.46 m or 45.64 m arise if one mistakenly uses 20 instead of (H − 20) or misuses the 60° relation. 54.46 m is a slight rounding error in the wrong direction and does not match 20(√3 + 1). 40 m ignores the trigonometric relations entirely and is much too small given that the building itself is already 20 m high and the tower must be taller than the building. Common Pitfalls: Students often confuse angle of elevation and depression or forget that the angle of depression equals the angle of elevation from the foot of the tower to the building top. Another common error is to use 20 instead of (H − 20) for the vertical side corresponding to the 60° angle. Carefully distinguishing the height difference is crucial. Final Answer: The height of the tower is approximately 54.64 m.

Question 15

In a right-triangle situation, a ladder is leaning against a perfectly vertical wall and makes an angle of elevation of 60° with the horizontal ground. If the foot of the ladder is placed 4.6 m away from the wall, what is the length of the ladder in…

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Introduction / Context: This height and distance question applies basic trigonometry to a real-life scenario where a ladder leans against a wall. You are expected to use the angle of elevation and the horizontal distance from the wall to determine the actual length of the ladder, which forms the hypotenuse of a right triangle. Given Data / Assumptions: The wall is perfectly vertical and the ground is perfectly horizontal, so the ladder, wall and ground form a right triangle. Angle of elevation between the ladder and the ground = 60°. Horizontal distance between the foot of the ladder and the wall = 4.6 m. We must find the length of the ladder in metres. Concept / Approach: When a ladder leans against a wall, the ladder is the hypotenuse of a right triangle, the distance from the wall is the adjacent side, and the height of contact with the wall is the opposite side. The cosine of the angle of elevation is defined as adjacent / hypotenuse. Therefore, we can use the cosine formula: cos(theta) = adjacent / hypotenuseand rearrange it to solve for the hypotenuse (the ladder length). Step-by-Step Solution: Let L be the length of the ladder.Angle of elevation, theta = 60°.Adjacent side (distance from wall) = 4.6 m.cos(60°) = adjacent / hypotenuse = 4.6 / LWe know cos(60°) = 1 / 2.So, 1 / 2 = 4.6 / LCross-multiplying gives: L = 4.6 * 2 = 9.2Therefore, the ladder length L = 9.2 m. Verification / Alternative check: We can quickly check using the sine relation. The vertical height h would be: h = L * sin(60°) = 9.2 * (√3 / 2)This gives a positive realistic height value, confirming that the calculated hypotenuse is reasonable for the given horizontal distance of 4.6 m and angle 60°. Why Other Options Are Wrong: 7.8 m: Too short; with a 60° angle and base 4.6 m it would imply cos(60°) ≠ 4.6 / 7.8.4.6 m: This would correspond to a 60° angle only if the ladder were equal to the base, which contradicts the cosine relationship.2.3 m: Much too short; the ladder could not even reach the wall at any realistic angle.8.4 m: Closer, but still inconsistent with cos(60°) = 4.6 / L. Common Pitfalls: Students often confuse sine and cosine and might incorrectly use sin(60°) = opposite / hypotenuse with the base instead of the height. Another common mistake is to assume the ladder length equals the base distance. Also, forgetting that cos(60°) = 1 / 2 can lead to incorrect numerical values. Final Answer: The length of the ladder is 9.2 m.

Question 16

The upper part of a tree breaks at a certain height and falls so that the broken top touches the ground, making an angle of 60° with the horizontal at a point 10 m away from the foot of the tree. What was the original height of the tree in metres?

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Introduction / Context: This problem is a classic height-and-distance application involving a broken tree. The tree originally stood vertically, then broke at some height, and its top touched the ground forming a right triangle with the tree trunk and the ground. We use trigonometric ratios to determine the original height of the tree. Given Data / Assumptions: The tree was originally vertical. The top portion broke and now touches the ground. The broken top makes an angle of 60° with the ground. The horizontal distance from the tree's foot to where the top touches the ground is 10 m. We ignore the thickness of the tree and assume it is a straight line segment. Concept / Approach: Once the tree breaks, the upper part becomes the hypotenuse of a right triangle whose base is the distance from the tree foot to the point where the top touches the ground. The angle between this hypotenuse and the ground is 60°. The vertical segment from the ground to the breaking point is one leg of the triangle, and the hypotenuse is the length of the broken part. The original height equals the sum of the stump height plus the length of the broken part (since that broken piece used to be vertical as well). Step-by-Step Solution: Let L be the length of the broken top (hypotenuse).Base (distance along ground) = 10 m.Angle with ground = 60°.cos(60°) = base / hypotenuse = 10 / Lcos(60°) = 1 / 2, so 1 / 2 = 10 / LTherefore, L = 10 * 2 = 20 m.Height of breaking point above ground = L * sin(60°)sin(60°) = √3 / 2, so stump height = 20 * (√3 / 2) = 10√3 m.Original height of tree = stump height + broken part length = 10√3 + 20.Factorizing: 10√3 + 20 = 10(√3 + 2) = 10(2 + √3) m. Verification / Alternative check: We can verify by reconstructing: if original height is 10(2 + √3), then the part above the breaking point is 20 m. Placing a 20 m segment at 60° with the ground gives a horizontal projection of 20 * cos(60°) = 10 m, which matches the given distance. The vertical projection 20 * sin(60°) equals 10√3 m, which is the stump height. Everything is consistent. Why Other Options Are Wrong: 10√3 m: This is only the stump height, not the total height.20√3 m: Overestimates and does not match the trigonometric relationships.10(2 - √3) m: This is much smaller and would imply an unrealistic geometry.15(2 + √3) m: Arbitrary multiple that does not satisfy the given 10 m base condition. Common Pitfalls: Many learners forget that the original height is the stump plus the broken portion, not just one of them. Another frequent error is using tan instead of cos for the base, or misusing √3/2 and 1/2 for 60°. Carefully identifying which side is adjacent, opposite, and hypotenuse relative to the given angle is essential. Final Answer: The original height of the tree is 10(2 + √3) m.

Question 17

A boat moves directly away from the foot of an observation tower. At one instant, the angle of depression of the boat from the observer's eye at the top of the tower is 60° when the boat is 50 m from the tower. After 8 seconds, the angle of depressi…

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Introduction / Context: This question tests your understanding of angles of depression and how to convert a geometric change in position into a speed in km/h. The observer stands at the top of a tower, and as the boat moves away, the angle of depression decreases from 60° to 30° over a known time interval. Given Data / Assumptions: Angle of depression to the boat initially = 60°. Horizontal distance of the boat from the tower at that instant = 50 m. Angle of depression after 8 seconds = 30°. Boat moves in a straight horizontal line directly away from the tower. The height of the tower remains constant. We assume no currents and treat water as still, so motion is uniform. Concept / Approach: An angle of depression from the top of a tower equals the angle of elevation from the boat to the top of the tower. Using basic trigonometry, tan(theta) = opposite / adjacent, where the opposite side is the tower height and the adjacent side is the horizontal distance. We can find the height of the tower from the first observation, use it to determine the second distance, then compute how far the boat travelled in 8 seconds and convert that to km/h. Step-by-Step Solution: Let h be the height of the tower.Initially, tan(60°) = h / 50.tan(60°) = √3, so h = 50 * √3.After 8 s, angle of depression = 30°, so tan(30°) = h / d, where d is new distance.tan(30°) = 1 / √3, hence 1 / √3 = (50 * √3) / d.So, d = 150 m.Distance travelled in 8 s = d - 50 = 150 - 50 = 100 m.Speed in m/s = 100 / 8 = 12.5 m/s.Convert to km/h: speed = 12.5 * 3.6 = 45 km/h. Verification / Alternative check: If the speed is 45 km/h, then per second the boat travels 45 * 1000 / 3600 = 12.5 m/s, confirming the displacement of 100 m in 8 seconds. The resulting distances and angles are consistent with the tangent values for 60° and 30°. Why Other Options Are Wrong: 33 km/h and 36 km/h: These represent smaller speeds and would not allow the boat to move 100 m in just 8 seconds.42 km/h: Close but slightly low; gives less than 100 m in 8 seconds.50 km/h: Slightly high; would imply a greater horizontal displacement than allowed by the angle change. Common Pitfalls: Common mistakes include using the sine instead of the tangent function, forgetting that angle of depression equals angle of elevation, or failing to convert from m/s to km/h correctly (multiplying by 3.6). Another pitfall is forgetting to subtract the initial distance to find the distance actually travelled in the 8 seconds. Final Answer: The approximate speed of the boat is 45 km/h.

Question 18

From a point on the ground, the angle of elevation of an aeroplane is initially 45°. After the aeroplane flies horizontally for 15 seconds, the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 2500 metres above the …

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Introduction / Context: This is a classic trigonometry and relative motion problem. The aeroplane flies horizontally at a constant height, causing the angle of elevation at a fixed observation point on the ground to change from 45° to 30° over a known time. From this change in viewing geometry, we deduce the horizontal speed of the aeroplane in km/hr. Given Data / Assumptions: Initial angle of elevation of the aeroplane = 45°. Final angle of elevation after 15 seconds = 30°. Constant height of the aeroplane above the ground = 2500 m. The aeroplane moves in a straight horizontal line. The observer is fixed at one point on level ground. We ignore effects like Earth curvature and assume right-triangle geometry holds. Concept / Approach: For a fixed height h and horizontal distance x from the observer, we use the tangent relation: tan(theta) = h / xAt 45° and 30°, the corresponding distances x1 and x2 can be computed. The change in these distances over 15 seconds gives the horizontal speed. Finally, we convert the speed from m/s to km/hr using the factor 3.6. Step-by-Step Solution: Let h = 2500 m.At first, tan(45°) = h / x1 ⇒ 1 = 2500 / x1 ⇒ x1 = 2500 m.After 15 s, tan(30°) = h / x2 ⇒ 1 / √3 = 2500 / x2.So x2 = 2500 * √3 m.Horizontal distance travelled = x2 − x1 = 2500(√3 − 1) m.Time taken = 15 s.Speed in m/s = 2500(√3 − 1) / 15.Multiply by 3.6 to convert to km/hr:Speed = [2500(√3 − 1) / 15] * 3.6 = 600(√3 − 1) km/hr. Verification / Alternative check: We can approximate √3 ≈ 1.732. Then: Speed ≈ 600 * (1.732 − 1) = 600 * 0.732 ≈ 439.2 km/hr.This is a realistic cruising speed for an aeroplane, confirming that the formula and approach are plausible for the scenario. Why Other Options Are Wrong: 600 km/hr: Ignores the trigonometric adjustment due to the angles and uses an arbitrary value.600(√3 + 1) km/hr and 600 √3 km/hr: These are much larger and do not follow from the distance difference x2 − x1.500(√3 − 1) km/hr: Uses the correct functional form but with an incorrect coefficient due to wrong distance or time handling. Common Pitfalls: Students sometimes subtract angles (45° − 30°) and attempt to use a single tangent instead of working with two separate triangles. Another common error is to treat height as changing or to forget to convert from metres per second to kilometres per hour. Careful identification of x1 and x2 and proper time conversion are crucial. Final Answer: The speed of the aeroplane is 600(√3 − 1) km/hr.

Question 19

The shadow of a vertical tower, when the angle of elevation of the sun is 45°, is found to be 10 metres longer than the shadow of the same tower when the angle of elevation of the sun is 60°. What is the height of the tower in metres?

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Introduction / Context: This question connects the height of a vertical tower with the lengths of its shadows at two different angles of elevation of the sun, 45° and 60°. Using right-triangle trigonometry and the relation between tangent, height and shadow length, we can form an equation based on the given 10 m difference in shadow lengths and solve for the tower height. Given Data / Assumptions: The tower stands vertically on level ground. Angle of elevation of the sun in case 1 = 45°. Angle of elevation of the sun in case 2 = 60°. Shadow at 45° is 10 m longer than shadow at 60°. We are asked to find the tower height. Concept / Approach: For a vertical object of height h casting a shadow of length s, with angle of elevation theta, we have: tan(theta) = h / s ⇒ s = h / tan(theta)We compute the shadows for 45° and 60° in terms of h and then use the fact that the first shadow is 10 m longer than the second, leading to a simple equation in h. Step-by-Step Solution: Let h be the height of the tower.Shadow when angle is 45°: s1 = h / tan(45°) = h / 1 = h.Shadow when angle is 60°: s2 = h / tan(60°) = h / √3.Given: s1 is 10 m longer than s2 ⇒ s1 = s2 + 10.So, h = h / √3 + 10.Rearrange: h - h / √3 = 10.h(1 - 1 / √3) = 10.1 - 1 / √3 = (√3 - 1) / √3, so:h * (√3 - 1) / √3 = 10 ⇒ h = 10√3 / (√3 - 1).Multiply numerator and denominator by (√3 + 1):h = 10√3(√3 + 1) / (3 - 1) = 5√3(√3 + 1).Since √3 * (√3 + 1) = 3 + √3, we get h = 5(3 + √3) m. Verification / Alternative check: Approximate √3 ≈ 1.732. Then: h ≈ 5(3 + 1.732) = 5 * 4.732 ≈ 23.66 m.Shadow at 45°: s1 ≈ 23.66 m.Shadow at 60°: s2 ≈ h / √3 ≈ 23.66 / 1.732 ≈ 13.66 m.Difference s1 − s2 ≈ 10 m, matching the given condition. Why Other Options Are Wrong: 5(√3 - 1) m and 15(√3 - 1) m: These give much smaller heights and hence smaller shadows, not satisfying a 10 m difference.10(√3 - 1) m and 10(√3 + 1) m: When substituted, the shadow difference is not 10 m; the numbers do not fit the tangent relationships. Common Pitfalls: Students sometimes set up the equation as s2 = s1 + 10, reversing the longer and shorter shadow. Others may misuse tan(60°) as 1 / √3 instead of √3, which completely flips the equation. Another common error is failing to simplify fractions involving √3 properly when isolating h. Final Answer: The height of the tower is 5(3 + √3) m.

Question 20

Two men stand on opposite sides of a vertical tower. One measures the angle of elevation of the top of the tower as 30°, and the other measures the angle of elevation as 45°. If the height of the tower is 50 m, what is the distance between the two m…

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Introduction / Context: This question applies trigonometry to a situation where two observers stand on opposite sides of a tower and view its top at different angles of elevation. You must use the tangent ratio to find each observer's distance from the tower and then add them to obtain the separation between the two men. Given Data / Assumptions: The tower is vertical and 50 m high. Man A sees the top of the tower at an angle of elevation of 30°. Man B, on the opposite side, sees the top at an angle of 45°. Both men are assumed to stand on level ground at the same horizontal level as the base of the tower. The tower, both observers and the horizontal ground form right triangles. Concept / Approach: For each observer, the relationship between the tower height h and horizontal distance d from the tower is: tan(theta) = h / d ⇒ d = h / tan(theta)We compute distances d1 and d2 for 30° and 45° respectively, then sum them because the observers are on opposite sides of the tower in a straight line. Step-by-Step Solution: Height h = 50 m.For the observer at 30°: tan(30°) = 1 / √3.So, d1 = h / tan(30°) = 50 / (1 / √3) = 50√3.Using √3 ≈ 1.73, d1 ≈ 50 * 1.73 = 86.5 m.For the observer at 45°: tan(45°) = 1.So, d2 = h / tan(45°) = 50 / 1 = 50 m.The men are on opposite sides of the tower, so distance between them = d1 + d2.Total distance ≈ 86.5 + 50 = 136.5 m. Verification / Alternative check: We can quickly check that the approximate value matches the symbolic sum 50√3 + 50 = 50(√3 + 1). With √3 ≈ 1.73, this becomes 50 * 2.73 = 136.5 m, exactly the value used above. So our computations are consistent. Why Other Options Are Wrong: 50√3 m (~86.6 m): This is only the distance of the 30° observer from the tower, not the distance between the two men.100√3 m (~173.2 m): Overestimates the distance; comes from mis-adding or doubling 50√3.135.5 m: Close but slightly off; likely due to rounding or arithmetic error.86.6 m: Approximates only one side distance, not the separation. Common Pitfalls: Students sometimes subtract the distances instead of adding them because they misinterpret “opposite sides.” Another error is using tan(theta) = d / h instead of h / d. Also, not applying the given approximation √3 = 1.73 carefully can produce minor numerical discrepancies. Final Answer: The distance between the two men is approximately 136.5 m.

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