Chapter 8.2, Problem 46AYU

To determine

How much time was lost due to the error?

Answer to Problem 46AYU

  $\boxed{0.01hours}$

Explanation of Solution

Given information:

In attempting to fly from city $P$ to city $Q$ , an aircraft followed a course that was $10°$ in error, as indicated in the figure. After flying a distance of $50miles$ , the pilot corrected the course by turning at point $R$ and flying $70miles$ farther. If the constant speed of the aircraft was $250miles$ per hour, how much time was lost due to the error?

  

Calculation:

Use the law of sine to find the measure of angle $Q$ .

  $\begin{array}{l}\frac{\sin P}{P}=\frac{\sin Q}{Q}\\ \frac{\sin 10°}{70}=\frac{\sin Q}{50}\\ now,\\ \sin Q=\frac{50\left(\sin 10°\right)}{70}\\ Q={\sin }^{-1}\frac{50\left(\sin 10°\right)}{70}\\ Q=7.12\end{array}$

  $\begin{array}{l}P+Q+R=180°\\ 10°+7.12°+R=180°\\ R=180°-17.12°\\ R=162.88°\end{array}$

Using law of sine, to find $r$

  $\begin{array}{l}\frac{\sin 10°}{70}=\frac{\sin 162.88°}{r}\\ r=\frac{\sin 162.88°}{\sin 10°}\\ r=118.67\end{array}$

The speed of aircraft $250miles/hour$

  $\begin{array}{l}time=\frac{dis\tan ce}{speed}\\ time=\frac{118.67}{250}\\ time=0.47hours\end{array}$

The total distance travelled by the aircraft is $\left(50+70\right)or120miles$

Time taken due to error is,

  $\begin{array}{l}\frac{120}{250}=0.48hours\\ 0.48-0.47=0.01hours\end{array}$

Hence, time lost due to error is, $\boxed{0.01hours}$