Question

A supermarket is to be designed as a rectangular building with a floor area of 14,400 square feet. The front and back of the building will be mostly glass and will cost $81 per running foot. The other two walls will be constructed of brick and cost $64 per running 1 foot. Find the dimensions of the base of the building that will minimize cost.

Answer 1

Answer:

• front/back: 106 2/3 ft
• sides: 135 ft

Step-by-step explanation:

These problems are easily solved if you start with the knowledge that the solution makes the front/back cost equal to the side cost.

Suppose we define the length of the front as x. Then the total cost of the front and back is (2x)(81) = 162x.

If y is the length of the side of the building, then (2y)(64) = 128y is the total cost of the sides of the building. When these costs are equal, we have ...

  162x = 128y

  y = (162/128)x

The floor area is ...

  xy = 14400 = x(162/128)x

  x = √(14400·128/162) = √(11377 7/9) = 106 2/3

  y = (162/128)x = 135

The front/back of the building measure 106 ft 8 inches; the sides measure 135 feet.

_____

Solution using derivatives

Using the above variable definitions, we can find the side length as ...

  y = 14400/x

so the total cost is then ...

  c = 162x + 128(14400/x)

We want the derivative with respect to x to be zero:

  dc/dx = 0 = 162 -128·14400/x^2

Solving for x gives ...

  x = √(14400·128/162) = 106 2/3 . . . . . compare to the solution above

  y = 14400/(106 2/3) = 135