Maximizing Area of Rectangle w/440yd Perimeter

[wcbryant87]

Homework Statement

A running track consists of a rectangle with a semicircle at each end. If the perimeter is to be exactly 440 yards, find the dimensions (x and r) that maximize the area of the rectangle. [Hint The perimeter is 2x + 2$$\pi$$r

Homework Equations

The Attempt at a Solution

Ok I attempted this twice and got the exact same answer, twice. Here is what I did.

First I set up the equation: 440 = 2x + 2$$\pi$$r

I then set up the equation: Area (total) = $$\pi$$r² + 2rx where x is the length of the side of the field (not counting the semicircles) and r is the radius.

I solved for x from the first equation and came up with x = 220 - $$\pi$$r

I then plugged the value of x into the second equation. Once I destrubuted it, I took the derivative and set it to zero and had 2r$$\pi$$ + 440 - 4$$\pi$$r = 0

Solving for r, I got 70.03. The answer in the back of the book is 110. What am I doing wrong?

I appreciate the help!

 

[tiny-tim]

Hi wcbryant87!

… maximize the area of the rectangle.

erm … wrong area!

 

[wcbryant87]

haha. wow. the 'aha' moment has hit me. thanks!