With these restictions and guidlines...What is the minimum cost?

Question

A rectangular box with an open top is to have a volume of 12 cubic meters. The length of the box is three times that of the width. The material to make the base consts $16 per squate meter and the material to make the side costs $8 per square meter. Find the dimensionsof the box that will minimize the cost. What is the minimum cost?Please give a step by step process if it is at all possible.

Answer 1

The volume of any rectangular box is length times width times height.

 

Let width = x

Let length = 3x

Let height = y

 

 

Knowing this, we can say that the base contains the length and width.  If we draw the open box and label it correctly, we will have 4 sides and one base.  This is important to know for the area.

 

Area of base = 3x(x) = 3x²

 

Total area of sides = 2xy + 2(3xy)

                           = 2xy + 6xy

                           = 8xy

 

 

Each type of area has it own cost.  We can set up equations using the information. Once for volume and the other for cost.

 

 

12 = 3x²y    ------>  volume

 

C = 16(3x²) + 8(8xy)

 

C = 48x² + 64xy         ------> cost

 

 

Substitute the volume equation into the cost equation to get cost equation in terms of x.

 

C = 48x² + 64x(4 / x²)

 

C = 48x² + 256x^-1

 

 

Now to find the dimensions that will minimize cost, we set the derivative of C equal to zero.

 

C' = 0

 

96x - 256x^-2 = 0

 

96x - (256 / x²) = 0

 

(96x³ - 256) / x² = 0

 

 

Set the numerator equal to zero and solve for x.  The x values will indicate possible maximum and minimum values.

 

96x³ - 256 = 0

 

96x³ = 256

 

    x³ = 2.67

 

    x = 1.38

 

Since we only have one x value, then the only dimensions that will minimize the cost is 1.38 meters.

 

To find the minimum cost, evaluate C(1.38).