If the garden is to be 1250 square feet, and the fence along the driveway costs $6 per foot while on the other three sides it costs only $2 per foot, find the dimensions that will minimize the cost.

Dimensions of rectangular garden:

x = 25 feet (sides along the driveway)

y = 50 feet

Step-by-step explanation:

Rectangular area is:

A (r) = x*y (1)

if we call x one the driveway side the cost of that side will be

6*x

The cost of the other side parallel to driveway side is 2*x and cost of the others two sides are 4*y

Total costs: C = 6*x + 2*x * 4*y (2)

From equation (1)

A (r) = 1250 = x*y ⇒⇒ y = 1250 / x

Plugging that value in equation (2) we get costs as a function of x

that is:

C (x) = 6*x + 2*x + 4 * 1250/x

Taking derivatives on both sides of the equation

C' (x) = 6 + 2 - 5000/x²

C' (x) = 8 - 5000 / x²

C' (x) = 0 ⇒ 8 - 5000 / x² = 0

8*x² - 5000 = 0

x² = 5000/8

x² = 625

x = 25 feet

and y = 1250 / 25

y = 50 ft

C (min) = 50*2*2 + 6*25 + 2*25

C (min) = 200 + 200

C (min) = 400 $