Chapter 9, Problem 32CR

To determine

To find: The longest distance across the pool, the shortest distance and the distance between the foci

Answer to Problem 32CR

The longest distance across the pool is 36 feet, the shortest distance is 28 feet and the distance between the foci is $16\sqrt{2}$ feet.

Explanation of Solution

Given information:

The equation for the elliptical shape of the pool that is measured in feet is $\frac{x^2}{324}+\frac{y^2}{196}=1$

Formula used:

The standard form of ellipse for $\left(0,0\right)$ center is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The formula to find foci is $c=\sqrt{a^2-b^2}$

Calculation:

The solution can be obtained as,

Here, $a=\sqrt{324}=18$ and $b=\sqrt{196}=14$ [According to the formula]

Now, $a\text{ and }b$ is identified and the value of $2a$ and $2b$ is obtained as,

  $2a=36\text{ and }2b=28$

Here $2a$ is the longest distance and $2b$ is the shortest distance

The value of $c$ can be obtained as,

  $\begin{array}{l}c=\sqrt{a^2-b^2}\\ c=\sqrt{{18}^2-{14}^2}\\ c=\sqrt{324-196}\\ c=\sqrt{129}\\ c=8\sqrt{2}\\ 2c=16\sqrt{2}\end{array}$

The distance between the foci is $2c$

Therefore, the longest distance across the pool is 36 feet, the shortest distance is 28 feet and the distance between the foci is $16\sqrt{2}$ feet.