Answer:
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)(-2L<sup>2</sup> + 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.
