Extreme Value Problem

Question

A Rectangular Box with a Volume of 931 ft cubed is to be constructed with a square base and top. The cost per square foot for the bottom is 20 cents, for the top 15 cents, and for the sides 1.5 cents. What dimensions will minimize the cost?

Answer 1

The rectangular box has 4 sides and 2 square bases.  Each pair of opposite sides have the same area.

 

Let length = x

Let width = x

Let height = y

 

Volume = 931 ft³

 

Surface area is the sum of the flat areas of the box and each area has a cost per square-foot.  If we draw our box and label it correctly, we can write equations.

 

x²y = 931

 

SA = 2x² + 4xy 

 

Since the cost is associated with area, we will focus on the surface area formula.  We need to write a formula for cost C using the area. Let's get the formula in terms of x.  C is in cents.

 

C = 20x² + 15x² + 1.5(4xy)

C = 35x² + 6xy

C = 35x² + 6x(931/x²)

C = 35x² + 5586x^-1

 

Now, we set the derivative of C equal to zero to find the minimum.

 

d/dx(C) = 0

 

70x - 5586x^-2 = 0

 

(70x³ - 5586) / x² = 0

 

 

Set the numerator part equal to zero.

 

70x³ - 5586 = 0

 

70x³ = 5586

 

x³ = 79.80

 

x = 4.30

 

Next, we will perform 2 test points around this value of x.  Use x=3    and   x=5.  Plug in these values into the derivative. 

 

C'(3) = 70(3)³ - 5586

        = -3696

 

C'(5) = 70(5)³ - 5586

        = 3164

 

The derivative decreases then increases.  This indicates a minimum.  We plug in the value of x into the dimensions.

 

length = 4.30 ft

width = 4.30 ft

height = 50.30 ft