Diana has available 200200 yards of fencing and wishes to enclose a rectangular area.
(a) Express the area A of the rectangle as a function of the width W of the rectangle.
(b) For what value of W is the area largest?
(c) What is the maximum area?
(a) A(W)=
To find the area A of the rectangle as a function of the width W, we first need to consider the perimeter of the rectangle. The perimeter of a rectangle is given by the formula:
Perimeter = 2*(length + width)
Given that the total length of fencing available is 200 yards, we can express the perimeter in terms of the width W as follows:
200 = 2*(W + length)
Solving for length, we get:
length = 100 - W
The area A of a rectangle is given by the formula:
Area = length * width
Substitute the expression for length in terms of W into the formula for area, we get:
A(W) = (100 - W) * W
A(W) = 100W - W^2
So, A(W) = 100W - W^2.
(b) To find the value of W for which the area is largest, we need to find the maximum point of the function A(W). Taking the derivative of A(W) and setting it equal to zero will give us the critical points:
A'(W) = 100 - 2W = 0
2W = 100
W = 50
Therefore, the width for which the area is largest is W = 50.
(c) To find the maximum area, we substitute W = 50 back into the area function:
A(50) = 100(50) - (50)^2
A(50) = 5000 - 2500
A(50) = 2500
Therefore, the maximum area is 2500 square yards.