Chapter 9, Problem 53RE

To determine

Find the distance of the ship that is facing towards north.

Answer to Problem 53RE

The distance of the ship is at the $72miles$ .

Explanation of Solution

Given:

It is given in the question that a radio transmitting station A is located $200$ miles east of transmitting station B. A ship is an area to the north and $40$ miles west of station A. Synchronized radio pulses transmitted to the ship at $186,000$ miles per second by the two stations are received $0.0005$ second sooner from station A than from station B.

Concept Used:

In this, use the concept of using the parabola and also use the difference between the pulses and also using the relationship $c^2=a^2+b^2$ .

Calculation:

Let the foci be $(100,0)$ for station A & $(-100,0)$ for station B so that the center is at the origin & the distance between $2$ foci is $2c=200$ miles. Also,

the ship is located $40$ miles west of station A or at the coordinates $(100-40,y)or(60,y)$ .

  $\begin{array}{l}2c=200\\ c=100\end{array}$

Since station A received the pulses $0.0005$ second sooner, then it was sent $186000(0.0005)=93$ miles farther from B than from A.

This difference also corresponds to the fixed distance $2a$ ,

  $\begin{array}{l}2a=93\\ a=46.5\end{array}$

Solving for $b^2$ using the relationship $c^2=a^2+b^2$

  $\begin{array}{l}{c}^2=a^2+b^2\\ b^2=c^2-a^2\Rightarrow {100}^2-{46.5}^2=7837.75\end{array}$

The equation of the parabola is:

  $\begin{array}{l}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \frac{x^2}{{46.5}^2}-\frac{y^2}{7837.75}=1\\ \frac{x^2}{2162.25}-\frac{y^2}{7837.75}=1\end{array}$

At $x=60$ , solve for y (ship’s location):

  $\begin{array}{l}\frac{{60}^2}{2162.25}-\frac{y^2}{7837.75}=1\\ \frac{y^2}{7837.75}=\frac{{60}^2}{2162.25}-1\\ y=\sqrt{7837.75(\frac{{60}^2}{2162.25}-1)}\\ y\approx 72miles\end{array}$

Conclusion:

The ship is at the $72miles$ .