Question

from a window 32.0 ft above the street, the angle of elevation to the top of the building across the street is 53.0° and the angle of depression to the base of this building is 17.0°. find the height of the building across the street. the height of the building across the street is \boxed{} ft (round to the nearest whole number as needed.)

Explanation:

Step1: Find the horizontal distance (let's call it \( x \)) between the window and the building using the angle of depression.

The angle of depression is \( 17.0^\circ \), and the vertical distance from the window to the street is \( 32.0 \) ft. We can use the tangent function: \( \tan(17.0^\circ) = \frac{32.0}{x} \), so \( x = \frac{32.0}{\tan(17.0^\circ)} \).
Calculating \( \tan(17.0^\circ) \approx 0.3057 \), then \( x \approx \frac{32.0}{0.3057} \approx 104.68 \) ft.

Step2: Find the vertical distance (let's call it \( y \)) from the window to the top of the building using the angle of elevation.

The angle of elevation is \( 53.0^\circ \), and the horizontal distance \( x \approx 104.68 \) ft. Using the tangent function again: \( \tan(53.0^\circ) = \frac{y}{x} \), so \( y = x \cdot \tan(53.0^\circ) \).
Calculating \( \tan(53.0^\circ) \approx 1.3270 \), then \( y \approx 104.68 \cdot 1.3270 \approx 138.91 \) ft.

Step3: Find the total height of the building.

The total height \( H \) is the sum of the height from the street to the window (\( 32.0 \) ft) and the height from the window to the top of the building (\( y \)): \( H = 32.0 + 138.91 \approx 170.91 \) ft. Rounding to the nearest whole number, we get \( 171 \) ft.

Answer:

\( 171 \)