Ernesto plans to build a pen for his pet iguana. Think about these two different questions. What is the area of the largest rectangular pen that he can make with 400 meters of fencing? What is the area of the largest circular pen he can make with the 400 meters of fencing?

The area of the rectangular pen is 10,000 m² and the area of the circular pen is 12,738.95 m².

For the rectangular pen, we want the sides to be congruent. To minimize perimeter and maximize area, the figure needs to be equilateral.

400/4 = 100 m for each side

This means that the area is 100 (100) = 10,000 m².

If the pen is circular, this means the circumference is 400 m. Circumference is given by the formula C=πd:

400 = 3.14d

Divide both sides by 3.14:

400/3.14 = 3.14d/3.14

127.389 = d

The radius is half of the diameter:

r = 127.389/2 = 63.6945

The area of the circle is given by A=πr²:

A=3.14 (63.6945) ² = 12738.9465 ≈ 12738.95 m²