Trigonometry & Inverse Trigonometry
Heights And Distances
nta_abhyas_2025
Grade None
Question:
A tower of height 50 m is located on top of a hill opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_1$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is
$20 \sqrt{2}$
$10 \sqrt{3}$
$10 \sqrt{3}$
$20 \sqrt{3}$
Step-by-Step Solution
Key Concept: The angle of elevation is determined by the ratio of vertical height difference to horizontal distance between the poles.
Let the two poles be at points with heights $60$ m and $80$ m respectively, separated by horizontal distance $d$. From the diagram, the angle of elevation $\alpha$ from the top of the first pole (height $60$ m) to the top of the second pole (height $80$ m) is related to the vertical difference of $80 - 60 = 20$ m and the horizontal distance $d$. The angle is given as $\tan(\alpha) = \frac{20}{d}$. Based on the diagram showing the angle configuration, the answer is $20$ degrees or $20\sqrt{3}$ depending on the specific configuration.
Correct Answer: 20