Question

Robert has available 400 yards of fencing and wishes to enclose a rectangular area. Express the areaAof the rectangle as a function of the widthwof the rectangle. For what value ofwis the arealargest? What is the maximum area?

Answer 1

The area of the rectangle can be expressed as A = w(400 - 2w). The maximum area occurs when the width of the rectangle is 100 yards, and the maximum area is 20000 square yards.

Step-by-step explanation:

To express the area A of the rectangle as a function of the width w, we can use the formula A = w(400 - 2w).

To find the value of w that gives the largest area, we can take the derivative of the area function with respect to w and set it equal to zero. By solving this equation, we can determine the value of w. The maximum area can then be found by substituting this value of w back into the area function.

The maximum area occurs when the width of the rectangle is 100 yards, and the maximum area is 20000 square yards.

Answer 2

A) A = 200w - w²

B) w = 100 yards

C) Max Area = 10000 sq.yards

Step-by-step explanation:

We are told that Robert has available 400 yards of fencing.

A) we want to find the expression of the area in terms of the width "w".

Since width is "w", and perimeter is 400,if we assume that length is l, then we have;

2(l + w) = 400

Divide both sides by 2 gives;

l + w = 200

l = 200 - w

Thus, Area of rectangle can be written as;

A = w(200 - w)

A = 200w - w²

B) To find the value of w for which the area is largest, we will differentiate the expression for the area and equate to zero.

Thus;

dA/dw = 200 - 2w

Equating to zero;

200 - 2w = 0

2w = 200

w = 200/2

w = 100 yards

C) Maximum area will occur at w = 100.

Thus;

A_max = 200(100) - 100(100)

A_max = 10000 sq.yards