A rectangular area of 36 f t2 is to be fenced off. Three sides will use fencing costing $1 per foot and the remaining side will use fencing costing $3 per foot. Find the dimensions of the rectangle of least cost. Make sure to use a careful calculus argument, including the argument that the dimensions you find do in fact result in the least cost (i. e. minimizes the cost function).

x = 8,49 ft

y = 4,24 ft

Step-by-step explanation:

Let x be the longer side of rectangle and y the shorter

Area of rectangle = 36 ft² 36 = x * y ⇒ y = 36/x

Perimeter of rectangle:

P = 2x + 2y for convinience we will write it as P = (2x + y) + y

C (x, y) = 1 * (2x + y) + 3 * y

The cost equation as function of x is:

C (x) = 2x + 36/x + 108/x

C (x) = 2x + 144/x

Taking derivatives on both sides of the equation

C' (x) = 2 - 144/x²

C' (x) = 0 2 - 144/x² = 0 ⇒ 2x² - 144 = 0 ⇒ x² = 72

x = 8,49 ft y = 36/8.49 y = 4,24 ft

How can we be sure that value will give us a minimun

We get second derivative

C' (x) = 2 - 144/x² ⇒C'' (x) = 2x (144) / x⁴

so C'' (x) > 0

condition for a minimum