Question

A woman entering an outside glass elevator on the ground floor of a hotel glances up to the top of the building across the street and notices that the angle of elevation is 48°. She rides the elevator up three floors (60 feet) and finds that the angle of elevation to the top of the building across the street is 31°. How tall is the building across the street? (Round to the nearest foot.)

Answer 1

131 ft

Explanation:

You want the height of a building if two observations of its top separated by 60 vertical feet have angles of elevation of 48° and 31°.

Setup

Let x be the distance to the building at street level, and y be the height of the building. The tangent relation is ...

Tan = Opposite/Adjacent

The first observation tells us ...

tan(48°) = y/x

The second observation tells us ...

tan(31°) = (y -60)/x

Solution

Solving each equation for x gives ...

x = y/tan(48°)

x = (y -60)/tan(31°)

The difference of these x-values is zero, so we have ...

x - x = 0

y/tan(48°) -(y -60)/tan(31°) = 0

y(1/tan(48°) -1/tan(31°)) +60/tan(31°) = 0

Subtracting the constant and dividing by the coefficient of y, we get ...

y = (60/tan(31°))/(1/tan(31°) -1/tan(48°)) = 60·tan(48°)/(tan(48°) -tan(31°))

y ≈ 130.724

The height of the building is about 131 feet.