Chapter SH, Problem 12.12EP

To determine

To find: The perimeter of the sail.

Answer to Problem 12.12EP

The perimeter of the sail is $96$ feet.

Explanation of Solution

Given:

The two angle measures are $40°$ and $55°$ , one side length measure is 25 feet.

Calculation:

Let the triangle is $\Delta ABC$ and two unknown sides are $x$ and $y$ as shown below.

  

Use the Law of Sines to find the missing sides.

  $\begin{array}{c}\frac{\sin 40°}{25}\approx \frac{\sin 55°}{x}\\ x\approx \frac{25\sin 55°}{\sin 40°}\\ x\approx 32\end{array}$

Find the angle $C$ .

  $\begin{array}{c}m\angle C=180°-\left(55°+40°\right)\\ =85°\end{array}$

Use the Law of Cosines $c^2=a^2+b^2-2ab\cos C$ to find the missing side length.

  $\begin{array}{c}{y}^2={32}^2+{25}^2-2\left(32\right)\left(25\right)\cos \left(85°\right)\\ =1509.550812\\ y\approx 39\end{array}$

Find the perimeter as follows.

  $\begin{array}{c}P=a+b+c\\ \approx 25+32+39\\ \approx 96\end{array}$

Therefore, the perimeter of the sail is $96$ feet.