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If the angle of elevation of the sun changes from 60° to 30°, the length of the shadow of a tower increases by 40 m. What is the height of the tower?
203m
403m
20 m
40 m
203m
Use the ratio 1:3 for base distances corresponding to 60° and 30°. The difference in ratio units (3−1/3) corresponds to 40 m, so h=3−140×3=203.
Angle of elevation changes from 60 to 30, increase in shadow length is 40 m.
h=cot(θ2)−cot(θ1)d
Use the ratio 1:3 for base distances corresponding to 60° and 30°. The difference in ratio units (3−1/3) corresponds to 40 m, so h=3−140×3=203.
Confusing the cotangent values or mistakenly subtracting the lengths without calculating the ratios relative to the height h.
Define relationships
Let h be the height of the tower. For θ=60°, the shadow length is x=h/3. For θ=30°, the shadow length is x+40=h3.
h3−3h=40
Simplify the equation
Multiply the equation by 3 to clear the fraction, resulting in 3h−h=403.
2h=403
Solve for h
Divide both sides by 2 to find the height h.
h=203
A is correct because the height of the tower calculated from the difference in shadow lengths is 203m.
This concept is fundamentally linked to vector resolution in Physics where you decompose height and distance components using similar trigonometric principles.
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