Question

Liana has 320 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum​ area? A rectangle that maximizes the enclosed area has a length of nothing yards and a width of nothing yards

Answer 1

Breadth = 60, Length = 60

Explanation

Let length & breadth of rectangular area be : L & B

As, Fencing Yards = 320. So, perimeter ie 2 (L + B) = 320

B = (320 - 2L)/2 → B = 160 - L (*)

Area of rectangle [A] = L x B = L (160 - L) → A = 160L - L^2

Maximising Area, So first derivative d [A] / d [L] = 160 - 2L

d [A] / d [L] ] = 0 → 160 - 2L = 0 → L = 160/2 → L = 80

By (*) : B = 160 - L = 160 - 80 → B = 80

Checking maximising condition, double derivative d^2[A] / d[L]^2 = -2

As d^2[A] / d[L]^2 is negative, L & B values are maximising A