Question

two office towers are 35m apart. from the top of the tower, the angle of elevation to the top of the taller tower is 20 degrees. from the top of the taller tower, the angle of depression to the bottom of the shorter tower is 61 degrees. determine the height of each building. include a diagram(s) with your solution

Answer 1

Explanations & Calculations

• Using the trigonometric identity $\small \tan\theta$, those needed quantities could be found.
• Refer to the attached figure

• From the triangle formed by the "top of the shorter tower, top of the taller tower, the gap between the buildings & the angle of elevation "

$\qquad\qquad \begin{aligned} \small \tan20&=\small \frac{h}{35}\\ \small h&=\small 35\times0.36397\\ &=\small \bold{12.74\,m} \end{aligned}$

• From the triangle formed by the "top of the taller tower, base of the shorter tower, gap between the buildings & the angle of depression"

$\qquad\qquad \begin{aligned} \small \tan61&= \small \frac{H}{35}\\ \small H &=\small 35\times1.80405\\ &=\small \bold{63.14\,m\,(\scriptsize height\,of\,the,taller\,building)} \end{aligned}$

• Then the height of the shorter building is

$\qquad\qquad \begin{aligned} &=\small H-h\\ &=\small \bold{50.40\,m} \end{aligned}$