Chapter 12.4, Problem 38PPS

To determine

To find: The measure of angle formed at Hilo.

Answer to Problem 38PPS

Distance between Captain cook and Kailua is $27.2mi$ .

Explanation of Solution

Given information:

Let Point at Hilo be H.

Point at Captain cook be C

Point at Kailua be K.

  $\begin{array}{l}C={80}^0\\ k=55mi\\ c=57mi\end{array}$

Given

  $\begin{array}{l}C={80}^0\\ k=55mi\\ c=57mi\end{array}$

Using law of sines

We get

  $\begin{array}{l}\frac{\sin C}{c}=\frac{\sin K}{k}\\ \Rightarrow \frac{\sin {80}^0}{57}=\frac{\sin K}{55}\\ \Rightarrow \sin K=\frac{55\sin {80}^0}{57}\\ \Rightarrow \sin K=\frac{55\left(0.9848\right)}{57}\\ \Rightarrow \sin K=\frac{54.164}{57}\\ \Rightarrow \sin K=0.95\\ \Rightarrow K={\sin }^{-1}\left(0.95\right)\\ \Rightarrow K={\sin }^{-1}\left(0.95\right)\\ \Rightarrow K={72}^0\end{array}$

We know

Sum of angles of a triangle is ${180}^0$

  $\begin{array}{l}\Rightarrow C+K+H={180}^0\\ \Rightarrow {80}^0+{72}^0+H={180}^0\\ \Rightarrow {152}^0+H={180}^0\\ \Rightarrow H={180}^0-{152}^0\\ \Rightarrow H={28}^0\end{array}$

Therefore,

Angle formed at Hilo is ${28}^0$

Using law of sines

We get

  $\begin{array}{l}\frac{\sin C}{c}=\frac{\sin H}{h}\\ \Rightarrow \frac{\sin {80}^0}{57}=\frac{\sin {28}^0}{h}\\ \Rightarrow h=\frac{57\sin {28}^0}{\sin {80}^0}\\ \Rightarrow h=\frac{57\left(0.47\right)}{0.9848}\\ \Rightarrow h=\frac{26.79}{0.9848}\\ \Rightarrow h=27.2\end{array}$

Therefore,

Distance between Captain cook and Kailua is $27.2mi$ .