Question

Suppose you still have 200 feet of fencing to enclose a rectangular garden. But this time to

maximize your area even more, you decide to use the side of a wall, so that only three sides
of the garden require fencing.
a. Find the function that will give you the maximum area.
b. Find the dimensions you should use that generate the largest possible area.
c. What is the largest possible area of your garden?
d. How much more area does using the side of a building add to your garden?

Answer 1

a. Area = (200 - 2*Width) * Width

b. Length = 100 feet and Width = 50 feet

c. Area = 5000 ft2

d. 2500 ft2 more area

Explanation:

The perimeter of a rectangle is given by: P = 2L + 2W, where L is the length and W is the width. The area of the rectangle is given by A = L * W.

As we will use one side of the rectangle as a wall, the perimeter of fence used will be:

P = L + 2W = 200

From this equation, we have L = 200 - 2W

Using this value of L in the area equation, we have:

A = (200 - 2W) * W = 200W - 2W^2

To find the maximum value of A, we need to find the vertix of the quadratic equation, and to do so we can use the following formula:

x_v = -b/2a = -200 / (-4) = 50

Using W = x_v = 50, we have that:

L = 200 - 2*50 = 100

A = L * W = 5000 ft2

If we didn't use the wall, the maximum area would be given by a square format, with length = 50 feet and width = 50 feet, giving the area of 50*50 = 2500 ft2

So the increase in the area was 5000 - 2500 = 2500 ft2