Question

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).

Answer 1

x = 8,49 ft

y = 4,24 ft

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