Chapter 9.2, Problem 57E

To determine

To find: The distance between the spark and the kidney stone.

Answer to Problem 57E

The distance between the spark and kidney stone is $187.62\text{mm}$ approximately.

Explanation of Solution

Given information:

A lithotripter machine uses an elliptical reflector to break up kidney stones nonsurgically. A spark plug in the reflector generates energy waves at one focus of an ellipse. The reflector directs these waves towards the kidney stone positioned at the other focus of the ellipse with enough energy to break up the stone as shown in the figure. The lengths of the major and minor axes of the ellipse are $260\text{mm}$ and $180\text{mm}$ , respectively.

  

Calculation:

As given, The lengths of the major and minor axes of the ellipse are $260\text{mm}$ and $180\text{mm}$ , respectively.

Use the formula for length of the major axis.

  $\begin{array}{l}2a=260\\ a=\frac{260}{2}\\ a=130\end{array}$

So, the value of $a$ is equal to $130\text{mm}$ .

Similarly Use the formula for length of the minor axis.

  $\begin{array}{l}2b=180\\ b=\frac{180}{2}\\ b=90\end{array}$

So, the value of $b$ is equal to $90\text{mm}$ .

Use the formula.

  $\begin{array}{l}{c}^2=a^2-b^2\\ c^2={130}^2-{90}^2\\ c^2=16900-8100\\ c^2=8800\\ c\approx 93.81\end{array}$

As per the formula the distance between the two foci is $2c$ .

So, the distance between the spark and kidney stone is $2\left(93.81\right)=187.62\text{mm}$ approximately.

Therefore, the distance between the spark and kidney stone is $187.62\text{mm}$ approximately.