Chapter 9, Problem 45CR

To determine

To calculate: The distance of the ship in the north

Answer to Problem 45CR

The ship is 72.2 miles far in the north

Explanation of Solution

Given information:

Radio transmitting station ‘A’ is located 200 miles east of station ‘B’

A ship is in an area north and 40 miles west of the station

A synchronized radio pulses transmitted to the ship at 186,000 miles per second by the two stations are received 0.005 second sooner from station ‘A’ than from station ‘B’

Calculation:

Use the difference between the pulses to find the value of $a,c\text{ and }b$

Here,

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

And,

  $\begin{array}{l}2a=\left(186000\right)\cdot \left(0.005\right)\\ 2a=93\\ a=46.5\end{array}$

As the value of $a\text{ and }c$ is obtained so the value of $b$ can be calculated as,

  $\begin{array}{l}b=\sqrt{c^2-a^2}\\ b=\sqrt{{100}^2-{\left(46.5\right)}^2}\\ b=\sqrt{7837.75}\\ b=88.53\end{array}$

The general equation of the parabola is

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

So, the equation of the parabola can be written as,

  $\begin{array}{l}\frac{x^2}{{46.5}^2}-\frac{y^2}{{88.53}^2}=1\\ \frac{x^2}{2162.5}-\frac{y^2}{7837.75}=1\end{array}$

The ship is 40 miles west from the station that means $x=100-40=60$

Hence the value of $y$ can be calculated as,

  $\begin{array}{l}{y}^2=b^2\left(\frac{x^2}{a^2}-1\right)\\ y^2=7837.75\left(\frac{{60}^2}{2162.5}-1\right)\\ y^2=7837.75\times 0.6647\\ y^2\approx 5211.57\\ y\approx 72.2\end{array}$

Therefore, the ship is 72.2 miles far in the north