Question

A 10-meter utility pole casts a 16-meter shadow directly down a slope when the angle of elevation of the sun is 42° (see figure). Find , the angle of elevation of the ground. (Round your answer to one decimal place.)

Answer 1

Solution:

Given the figure:

To find the angle θ, we have

step 1: Find angle A.

Thus, we have

$\begin{gathered} \angle\text{A+}\angle B+\angle C=180(sum\text{ of angles in a triangle\rparen} \\ \Rightarrow A+42+90=180 \\ A+132=180 \\ \Rightarrow A=180-132=48\degree \end{gathered}$

step 2: In the triangle ABC, solve for θ, using the sine rule.

Thus, we have

$\frac{\sin A}{x}=\frac{\sin B}{h}$

By substitution, we have

$\begin{gathered} \frac{\sin48}{16}=\frac{\sin(42-\theta)}{10} \\ this\text{ gives} \\ \sin(42-\theta)=\frac{10\times\sin48}{16} \\ \Rightarrow\sin(42-\theta)=0.46446 \\ take\text{ the sine inverse of both sides,} \\ \sin^{-1}(\sin(42-\theta))=\sin^{-1}(0.46446) \\ \Rightarrow42-\theta=27.68 \\ add\text{ -42 to both sides of the equation} \\ -42+42-\theta=-42+27.68 \\ \Rightarrow-\theta\text{=-14.32} \\ divide\text{ both sides by -1} \\ Hence, \\ \theta\approx\text{14.3 \lparen1 decimal place\rparen} \end{gathered}$

Hence, the angle of elevation of the ground, to 1 decimal place, is

$14.3\degree$