Question

Hello can u help? I need a solution to calculus problem I have $500.00 to spend on fencing for a garden. The fence for the street side costs $30.00 per foot and the other three sides cost $10.00 per foot. what dimensions will give you a rectangle for the largest possible area?
How do you know it is for max area?

Answer 1

with street side length x, and yard width y, then the cost is

c = 30x + 10(x+2y)
c=500, so

500 = 40x + 20y, so
y = 25 - 2x

the area is given by

a = xy = x(25-2x) = 25x - 2x^2
da/dx = 25-4x

da/dx=0 when x = 6.25. So, the yard is

6.25 x 12.5

The area has a max or min where da/dx = 0. Since a(x) is a parabola opening downward, it is a max.