Chapter 12.5, Problem 13P

To determine

Find x .

Answer to Problem 13P

  $x=\frac{9}{8}$

Explanation of Solution

Given:

The side of the beach shelter shown at the right is made up of four $30°-60°-90°$ triangles.

  

Formula Used:

  $\sin \theta =\frac{\text{length of opposite side to }\theta }{\text{length of hypotenuse }}$

Calculation:

In the figure , the lower angles are $30°$ and upper angles are $60°$ .

In the first white triangle,

  $\begin{array}{l}\sin 60°=\frac{\text{length of opposite side to 60}°}{\text{length of hypotenuse }}\\ \Rightarrow \sin 60°=\frac{a}{2}\\ \Rightarrow a=2\sin 60°\mathrm{...................}[\text{multiply each side by 2]}\\ \Rightarrow a=2\cdot \frac{\sqrt{3}}{2}=\sqrt{3}\end{array}$

In the next triangle,

  $\begin{array}{l}\sin 60°=\frac{\text{length of opposite side to 60}°}{\text{length of hypotenuse }}\\ \Rightarrow \sin 60°=\frac{b}{a}\\ \Rightarrow b=a\sin 60°\mathrm{...................}[\text{multiply each side by a]}\\ \Rightarrow b=\sqrt{3}\cdot \frac{\sqrt{3}}{2}=\frac{3}{2}\end{array}$

In the next triangle,

  $\begin{array}{l}\sin 60°=\frac{\text{length of opposite side to 60}°}{\text{length of hypotenuse }}\\ \Rightarrow \sin 60°=\frac{c}{b}\\ \Rightarrow c=b\sin 60°\mathrm{...................}[\text{multiply each side by b]}\\ \Rightarrow c=\frac{3}{2}\cdot \frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{4}\end{array}$

In the last triangle,

  $\begin{array}{l}\sin 60°=\frac{\text{length of opposite side to 60}°}{\text{length of hypotenuse }}\\ \Rightarrow \sin 60°=\frac{x}{c}\\ \Rightarrow x=c\sin 60°\mathrm{...................}[\text{multiply each side by c]}\\ \Rightarrow x=\frac{3\sqrt{3}}{4}\cdot \frac{\sqrt{3}}{2}=\frac{9}{8}\end{array}$

So, $x=\frac{9}{8}$