Perimeter (P) = 2L + 2w
64 = 2L + 2w
64 - 2L = 2w
32 - L = w
Area (A) = L x w
= L x (32 - L)
= 32L - L²
To find maximum area, calculate the derivative and set it equal to zero.
A' = 32 - 2L
0 = 32 - 2L
2L = 32
L = 16
Substitute L to solve for y: 32 - L = w → 32 - 16 = w → 16 = w
The maximum area will be 16 ft x 16 ft = 256 ft²