An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track. Find the dimensions that will produce a maximum area of the rectangular region.

The dimensions that will produce a maximum area of the rectangular region is 50 meters or 5000/Pi

Step-by-step explanation:

From the question given, let us recall the following formula

The perimeter = 200 which is,

P=2L + C

where L is = length of the rectangular region

The circumference of a circle denoted as C

The Circumference of the semi-circle is denoted as,

C=Pi x D, which is  D=C/Pi.

Thus the equation becomes,

200 = 2L+C

A=L x (C/Pi)

We now have Two equations and three variables, from these two equations, we can get a single equation,

200=2L+C means that C = 200-2L

For C in the Area equation. Substitute 200-2L:

A=Lx (C/Pi) which is  A = Lx (200-2L)/Pi.

We Simplify: A = (1/Pi)(-2L² + 200L)

Now take a  derivative of A with respect to L: dA/dL = (1/Pi)(-2L + 200)

(1/Pi)(-2L+200)=0

Let Solve for L: L = 50.

when L is 50 we have the MAXIMUM area. this is a negative quadratic so it MUST therefore not be a minimum but maximum

Then,

Plug in L=50 into the formula for A: A = 50(200-2(50))/Pi = 5000/Pi.