# A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is to be 1500 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

##### 1 Answer

The rectangle with maximum area should have dimensions

#### Explanation:

Assume the dimensions of a base rectangle are

Then the perimeter of a track equals to the lengths of two other sides (

The total perimeter is, therefore

(a)

We have to maximize the area of a rectangle, that is

(b)

From equation (a) we can derive

(c)

and substitute it into the formula (b) for area of rectangle:

This expression is a quadratic polynomial of

Having

CHECK against equation (a):