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------------------------------------------------ A sports field is to have the shape of a rectangle with semi-circles on the two ends. It must have a perimeter of 1200 meters. What is the maximum area possible for the rectangular part?

A. 30000 m²
B. 40000 m²
C. 45000 m²
D. 60000 m²

To find the maximum area of the rectangular part, we need to maximize the length and width of the rectangle while still maintaining a perimeter of 1200 meters.

Let the length of the rectangle be x meters.

The width of the rectangle would then be (600 - 2x) meters, because the total length of the two semicircles is equal to the width of the rectangle.

The area of a rectangle is given by length * width.

Substituting the expressions for length and width, the area becomes:

A = x * (600 - 2x) = 600x - 2x^2

The area is maximized when the derivative of A with respect to x is zero.

Differentiating and setting the derivative equal to zero, we get:

600 - 4x = 0

Solving for x, we find x = 150.

Plugging x = 150 into the area equation, we find the maximum area to be:

A = 150 * (600 - 2 * 150) = 30000 square meters.