We are going to fence in a rectangular field . Starting at the bottom of the field and moving around the field in a counter clockwise

Question

manner the cost of material for each side is $ 6/ft , 9/ ft , $12/ft and $14/ft respectively. if we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area.

Answer 1

Let x and y be the dimensions of the rectangle

Cost, C = 6x + 9y + 12x + 14y = 18x+ 23y = 1000

y = (1000-18x)/23

Area, A = xy

A(x) = x(1000-18x)/23 = (1000/23)x-18x²/23

Take the derivative of A(x) and then set that to zero to solve for x

A'(x) = 1000/23-36/23x

x = 1000/36 ≈27.78

y = (1000-18(27.78))/23 = 500/23 ≈ 21.74

The dimensions are 27.78 feet by 21.74 feet

The maximum area ≈ 604.91 feet