Question

You have 100 feet of fencing and decide to make a rectangular garden. What is the largest area enclosed

Answer 1

625 ft^2

Explanation:

Given

`P = 100` --- perimeter

Required

The largest area

The perimeter is calculated as:

`P = 2 * ( L + W)`

So, we have:

`2 * ( L + W) = 100`

Divide both sides by 2

`L + W = 50`

Make L the subject

`L = 50 - W`

The area is calculated as:

`A= L * W`

Substitute
`L = 50 - W`

`A= (50 - W) * W`

Open bracket

`A = 50W - W^2`

Differentiate with respect to W

`A' = 50 -2W`

Set to 0; to get the maximum value of W

`50 - 2W = 0`

Collect like terms

`-2W = -50`

Divide by -2

`W = 25`

So, the maximum area is:

`A = 50W - W^2`

`A = 50 * 25 - 25^2`

`A = 1250 - 625`

`A = 625`