A Surveyor's Trigonometry Challenge: Calculating the Height of a Radio Tower

A surveyor standing 68 meters from the base of a building measures the angle to the top of the building and finds it to be 36°. The surveyor then measures the angle to the top of the radio tower on the building and finds that it is 50°. How tall is the radio tower?

Final answer:

To find the height of the radio tower, we use trigonometry by calculating the heights to the building's top and the radio tower's top separately with the tangent function and then subtract the former from the latter.

Explanation:

The question involves finding the height of a radio tower on top of a building using trigonometry. Given that a surveyor is 68 meters from the base of the building and measures angles of elevation to the top of the building and the top of the radio tower as 36° and 50° respectively, we can use the tangent function to determine the heights.

Firstly, we find the height of the building using the formula height = distance × tan(angle). Thus, the height of the building (H1) is 68 × tan(36°). Then, using the same formula, we find the total height from the ground to the top of the radio tower (H2) which is 68 × tan(50°). The height of the radio tower itself is the difference between H2 and H1.

To find the height of the radio tower, we use trigonometry by calculating the heights to the building's top and the radio tower's top separately with the tangent function and then subtract the former from the latter.