Question

1. tanis lives on the 4th floor of her apartment building. from her window, the angle of depression is 38° to see the base of the neighbouring building. the distance between the buildings is 75 m. the angle of elevation is 62° to see the top of the neighbouring building. calculate the height of the neighbouring building. (2 marks)

Explanation:

Step1: Find the vertical distance from Tanis's position to the top - of - the - neighboring building

We know the angle of elevation $\theta = 62^{\circ}$ and the horizontal distance $d = 75$ m. We use the tangent function in a right - triangle $\tan\theta=\frac{h_1}{d}$, where $h_1$ is the vertical distance from Tanis's position to the top of the neighboring building. So $h_1 = d\times\tan\theta$.
$h_1=75\times\tan(62^{\circ})\approx75\times1.88073 = 141.05475$ m.

Step2: Find the vertical distance from Tanis's position to the base of the neighboring building

The angle of depression to the base of the neighboring building is $38^{\circ}$. Using the tangent function again in the right - triangle formed by Tanis's position, the horizontal line, and the line of sight to the base of the building. Let $h_2$ be the vertical distance from Tanis's position to the base of the neighboring building. $\tan(38^{\circ})=\frac{h_2}{d}$, so $h_2 = d\times\tan(38^{\circ})\approx75\times0.78129 = 58.59675$ m.

Step3: Calculate the height of the neighboring building

The height of the neighboring building $H=h_1 + h_2$.
$H=75\times\tan(62^{\circ})+75\times\tan(38^{\circ})=75(\tan(62^{\circ})+\tan(38^{\circ}))\approx75\times(1.88073 + 0.78129)=75\times2.66202=199.6515\approx199.65$ m.

Answer:

The height of the neighboring building is approximately $199.65$ m.