Chapter 6.3, Problem 67E

(a)

To determine

To calculate: The area of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as a function of $a$, if the area of the ellipse is given by $A=\pi ab$, and $a+b=20$.

(b)

To determine

To calculate: The equation of an ellipse with an area of $264$ square centimeters.

(c)

To determine

To fill:The provided table using the equation $A=\pi a\left(20-a\right)$.Also, make a conjecture about the shape of the ellipse with maximum area. The provided table is,

$\begin{array}{ccccccc}a& 8& 9& 10& 11& 12& 13\\ A& & & & & & \end{array}$

(d)

To determine

To graph: The area function $A=\pi a\left(20-a\right)$ using a graphing utility, and also use the graph to support the conjecture in part $\left(c\right)$.