Chapter 6.1, Problem 52E

a.

To determine

Draw a diagram that represents the problem.

Answer to Problem 52E

  $\boxed{\angle B=82°}$

Explanation of Solution

Given information:

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of $12°$ with the horizontal. The flagpole’s shadow is $16$ meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is $20°$ .

Draw a diagram that represents the problem. Show the known quantities on the diagram and use a variable to indicate the height of the flagpole.

Calculation:

A diagram that represents the problem,

  

  

Assume that height of flagpole is $h$ and the distance $AB\text{'}=16meters$

The angle $A$ is ,

  $\begin{array}{l}A=90°-12°\\ =78°\end{array}$

Sun of angles of the triangle,

  $\begin{array}{l}A+B+C=180°\\ \angle B=180°-A-B\text{'}\\ =180°-78°-20°\\ \angle B=82°\end{array}$

Hence, angle $\boxed{\angle B=82°}$ .

b.

To determine

Write an equation that you can use to find the height of the flagpole.

Answer to Problem 52E

  $\boxed{h=16\left(\frac{\sin 20°}{\sin 82°}\right)}$

Explanation of Solution

Given information:

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of $12°$ with the horizontal. The flagpole’s shadow is $16$ meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is $20°$ .

Write an equation that you can use to find the height of the flagpole.

Calculation:

According to the laws of sine.

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{16}{\sin B}=\frac{h}{\sin B\text{'}}\\ \frac{16}{\sin 82°}=\frac{h}{\sin 20°}\\ h=16\left(\frac{\sin 20°}{\sin 82°}\right)\end{array}$

Hence, an equation to find the height of the flagpole is $\boxed{h=16\left(\frac{\sin 20°}{\sin 82°}\right)}$

c.

To determine

Find the height of the flagpole.

Answer to Problem 52E

  $\boxed{5.45}meters$

Explanation of Solution

Given information:

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of $12°$ with the horizontal. The flagpole’s shadow is $16$ meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is $20°$ .

Find the height of the flagpole.

Calculation:

According to equation of height,

  $h=16\left(\frac{\sin 20°}{\sin 82°}\right)$

  $\begin{array}{l}h=16\left(\frac{0.34}{0.99}\right)\\ h\approx 5.45\end{array}$

Hence, the height of the flagpole is $\boxed{5.45}meters$ .