Question

47 - 50. solve. show the equation you use. write your answer in statement form.

1. from a window 30.0 ft above the street, the angle of elevation to the top of the building across the street is 50.0° and the angle of depression to the base of this building 20.0°. find the height of the building across the street.

Explanation:

Step1: Find the horizontal distance between the two buildings

Let the horizontal distance between the two buildings be \(x\).
We know that for the angle of depression \(\theta = 20.0^{\circ}\), and the height of the window \(h_1=30.0\) ft.
Since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), and the angle of depression is equal to the angle of elevation from the base of the opposite building to the window.
\(\tan(20.0^{\circ})=\frac{30.0}{x}\)
\(x = \frac{30.0}{\tan(20.0^{\circ})}\)
Using a calculator, \(\tan(20.0^{\circ})\approx0.3640\)
\(x=\frac{30.0}{0.3640}\approx82.42\) ft

Step2: Find the height from the window level to the top of the building

Let the height from the window level to the top of the building be \(h_2\)
For the angle of elevation \(\alpha = 50.0^{\circ}\) and the horizontal distance \(x\approx82.42\) ft
\(\tan(50.0^{\circ})=\frac{h_2}{x}\)
\(h_2=x\tan(50.0^{\circ})\)
Since \(x\approx82.42\) ft and \(\tan(50.0^{\circ})\approx1.1917\)
\(h_2 = 82.42\times1.1917\approx98.2\) ft

Step3: Find the total height of the building

The total height of the building \(H=h_1 + h_2\)
\(H=30.0+98.2 = 128.2\) ft

Answer:

The height of the building across the street is \(128.2\) ft.