Question

erik has 2000 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed area.

Answer 1

To find the dimensions of the rectangle that maximize the enclosed area, we need to use the given information about the amount of fencing available, and apply optimization techniques.

Step-by-step explanation:

To find the dimensions of the rectangle that maximize the enclosed area, we need to use the given information about the amount of fencing available. Let's assume the length of the rectangle is x yards. In this case, the width will be (2000 - 2x) yards (since the total fencing length is 2000 yards and two lengths of the rectangle are used). The area of the rectangle is given by the equation:

A = x(2000 - 2x)

By expanding and simplifying the equation, we get:

A = 2000x - 2x^2

To find the dimensions that maximize the area, we take the derivative of the area equation with respect to x and set it equal to 0, then solve for x. Once we find the value of x, we can calculate the corresponding width and area of the rectangle.

Answer 2

It is to be noted that where 2000 yards of fencing is given, to enclose a rectangular area, Dimensions of the largest area is 500 yds * 500 yds

2x + 2y = 2000

x + y = 1000

y = 1000 - x

Area of the field is:

A = xy

Substituting 1000 - x in for y we have the area in terms of x. We can find dy/dx to find its maximum value. The function is an inverted parabola, so will have one absolute maximum where the slope is = 0.

A(x) = x * (1000 - x) = 1000x - x 2

A‘(x) = 1000 - 2x

1000 - 2x = 0

2x = 1000

x = 500

At x = 500, you will find maximum area.

y = 1000 - x = 1000 - 500 = 500

Dimensions of the largest area is 500 yds * 500 yds