Chapter 8.4, Problem 49PS

Solution

To Find : Equations of ellipses with vertical major axes that model the largest and smallest fields described and an inequality describing the possible areas of these fields.

The equation that models the largest field is $\frac{x^2}{{\left(77.5\right)}^2}+\frac{y^2}{{\left(92.5\right)}^2}=1$ and the equation that model the smallest field is $\frac{x^2}{{\left(55\right)}^2}+\frac{y^2}{{\left(67.5\right)}^2}=1$ and the inequality describing the possible areas of these fields is $11700\le A\le 22500$ .

Given information :

The playing field for Australian football is an ellipse that is between $135\text{ and }185$ meters long and between $110\text{ and }155$ meters wide.

Explanation :

From the given information, it is clear that the length of the major axis $2a$ lies between $135\text{ and }185$ meters and the minor axis $2b$ lies between $110\text{ and }155$ meters.

The largest field will be the one with the largest major and minor axes lengths. So,

  $\begin{array}{c}2a=185\\ a=92.5\end{array}$

  $\begin{array}{c}2b=155\\ b=77.5\end{array}$

It is given that the ellipse has vertical major axes, so the equation of the largest field will be,

  $\begin{array}{c}\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\\ \frac{x^2}{{\left(77.5\right)}^2}+\frac{y^2}{{\left(92.5\right)}^2}=1\end{array}$

The smallest field will be the one with the smallest major and minor axes lengths. So,

  $\begin{array}{c}2a=135\\ a=67.5\end{array}$

  $\begin{array}{c}2b=110\\ b=55\end{array}$

It is given that the ellipse has vertical major axes, so the equation of the largest field will be,

  $\begin{array}{c}\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\\ \frac{x^2}{{\left(55\right)}^2}+\frac{y^2}{{\left(67.5\right)}^2}=1\end{array}$

The area of an ellipse is $\pi ab$ .

Area $\left(A\right)$ of the largest field will be,

  $\begin{array}{l}=3.14\times 77.5\times 92.5\\ \approx 22500\end{array}$

Area $\left(A\right)$ of the smallest field will be,

  $\begin{array}{l}=3.14\times 67.5\times 55\\ \approx 11700\end{array}$

So, the inequality representing the areas of these fields will be $11700\le A\le 22500$ .

Therefore, the equation that model the largest field is $\frac{x^2}{{\left(77.5\right)}^2}+\frac{y^2}{{\left(92.5\right)}^2}=1$ and the equation that model the smallest field is $\frac{x^2}{{\left(55\right)}^2}+\frac{y^2}{{\left(67.5\right)}^2}=1$ and the inequality describing the possible areas of these fields is $11700\le A\le 22500$ .