Chapter 8.2, Problem 37AYU

a

To determine

How far is each station from the ship?

Answer to Problem 37AYU

  $\boxed{135.58miles,143.33miles}$

Explanation of Solution

Given information:

Coast Guard Station Able is located $150$ miles due soth of station Baker(S.B). A ship at sea sends an SOS call that is received by each station. The call to station Able (S.A) indicates that the ship is located $N55°E$ ; the call to Station Baker indicates that the ship is located $S60°E$ .

How far is each station from the ship?

Calculation:

Consider the figure as per statement.

  

The sum of angles of triangle is $180°$ .

  $A+B+C=180°$

Substitute $60°$ for B and $60°$ for A we get.

  $\begin{array}{l}A+B+C=180°\\ A+60°+55°=180°\\ A=180°-115°\\ A=65°\end{array}$

Point C is the distance from station Able to ship,

Now apply law of sines,

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin C}{c}\\ \frac{\sin 65°}{150}=\frac{\sin 55°}{c}\\ 0.006=\frac{\sin 55°}{c}\\ c\left(0.006\right)=\sin 55°\\ c=\frac{\sin 55°}{0.006}\\ c=135.58miles\end{array}$

Point b is the distance from station Baker to ship,

Now apply law of sines,

  $\begin{array}{l}\frac{\sin 65°}{150}=\frac{\sin 60°}{b}\\ 0.006=\frac{\sin 60°}{b}\\ b\left(0.006\right)=\sin 60°\\ b=\frac{\sin 60°}{0.006}\\ b=143.33miles\end{array}$

Hence, the distance between stations is $\boxed{135.58miles,143.33miles}$ .

b.

To determine

How long will it take to reach the ship?

Answer to Problem 37AYU

  $\boxed{41\min utes}$

Explanation of Solution

Given information:

Coast Guard Station Able is located $150$ miles due south of station Baker (S.B). A ship at sea sends an SOS call that is received by each station. The call to station Able (S.A) indicates that the ship is located $N55°E$ ; the call to Station Baker indicates that the ship is located $S60°E$ .

If a helicopter capable of flying $200miles$ per hour is dispatches from the station nearest the ship, how long will it take to reach the ship?

Calculation:

The time taken for the helicopter to reach the ship can be determine,

  $time=\frac{distamce,traveled}{speed}$

The helicopter capable of flying $200miles$ per hour is dispatches from the station nearest the ship are.

  $\begin{array}{l}time=\frac{135.58}{200}\\ time=0.6779hours\end{array}$

Hence,the time taken by helicopter from nearest station is, $\boxed{41\min utes}$