Answer:
625 ft^2
Step-by-step explanation:
Given
[tex]P = 100[/tex] --- perimeter
Required
The largest area
The perimeter is calculated as:
[tex]P = 2 * ( L + W)[/tex]
So, we have:
[tex]2 * ( L + W) = 100[/tex]
Divide both sides by 2
[tex]L + W = 50[/tex]
Make L the subject
[tex]L = 50 - W[/tex]
The area is calculated as:
[tex]A= L * W[/tex]
Substitute [tex]L = 50 - W[/tex]
[tex]A= (50 - W) * W[/tex]
Open bracket
[tex]A = 50W - W^2[/tex]
Differentiate with respect to W
[tex]A' = 50 -2W[/tex]
Set to 0; to get the maximum value of W
[tex]50 - 2W = 0[/tex]
Collect like terms
[tex]-2W = -50[/tex]
Divide by -2
[tex]W = 25[/tex]
So, the maximum area is:
[tex]A = 50W - W^2[/tex]
[tex]A = 50 * 25 - 25^2[/tex]
[tex]A = 1250 - 625[/tex]
[tex]A = 625[/tex]
