Question

1. from the window of one building, samantha looks across the courtyard to a second building. samantha finds the angle of elevation to the top of the second building is 38° and the angle of depression to the bottom is 58°. if the buildings are 50 m apart, determine the height of the taller building correct to one decimal place. (2 marks)

Explanation:

Step1: Calculate the height from the window to the top

Let the height from the window to the top of the taller building be \(h_1\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Given \(\theta = 38^{\circ}\) and adjacent side \(= 50\)m.
Using the formula \(\tan38^{\circ}=\frac{h_1}{50}\), so \(h_1 = 50\times\tan38^{\circ}\).
Since \(\tan38^{\circ}\approx0.781\), then \(h_1=50\times0.781 = 39.05\)m.

Step2: Calculate the height from the window to the bottom

Let the height from the window to the bottom of the taller building be \(h_2\).
Given \(\theta = 58^{\circ}\) and adjacent side \(= 50\)m.
Using the formula \(\tan58^{\circ}=\frac{h_2}{50}\), so \(h_2 = 50\times\tan58^{\circ}\).
Since \(\tan58^{\circ}\approx1.600\), then \(h_2=50\times1.600=80\)m.

Step3: Calculate the total height of the taller building

The total height \(H\) of the taller building is \(H=h_1 + h_2\).
\(H=39.05+80=119.05\approx119.1\)m.

Answer:

\(119.1\)m