optimization problem help!

Question

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $4 per square foot. Find the dimensions that minimize cost if the box has a volume of 20 cubic feet.

Length of base x=                
Height of side z=                 

Answer 1

We must minimize C(x,z) = 5·2x² + 4·4xz = 10x² + 16xz under the constraint V = x²z = 20.

 

Method 1:

 

We eliminate z by substituting z = 20/x² into the cost.

 

C(x) = 10x² + 320/x

 

dC/dx = 20x - 320/x² = 0

 

20x³ = 320

 

x³ = 16

 

x = 2· ³√2

 

z = 20/x² = 5 / ³√4 

 

We apply the second derivative test

 

d²C/dx² = 20 + 640/x³

 

C''(2· ³√2) = 20 + 640/16 = 60 > 0

 

So this gives the minimum cost.

 

 

Method 2: Lagrange multipliers.

 

We solve the system

 

1:     ∂C/∂x = λ∂V/∂x

2:     ∂C/∂z = λ∂V/∂z

3:     V(x,z) = 20

 

This system is

 

1:    20x + 16z = 2λxz

2:    16x = λx²

3:     x²z = 20

 

Equation 2 gives us that either x = 0, which we discard (Volume would be 0), or λx = 16

 

Substituting λx = 16 into equation 1 and solving it for z gives z = 5x/4.

 

Plugging it into equation 3 gives

 

5x³/4 = 20

 

x³ = 16

 

x = 2· ³√2
z = 20/x² = 5 / ³√4

 

Since this is the only extremum, we only need to pick two points one with x < 2· ³√2 and one with x > 2· ³√2 and see that the cost is bigger than at the candidate point in both cases. This is the case here, so it is a minimum.

 

Another way is using a Bordered Hessian Criterion, but that would be harder.

Answer 2

Hi Brittany,

 

I will get you started on this one too.

 

We are looking for a cost function that depends on the length of the sides of the box.

 

Let the length of the square (top and bottom) be represented by x. Let the height of the box be represented by y.

 

The top of the box has an area of x² sq. ft. But each square foot costs $5. So the top costs 5x² . So does the bottom of the box.

 

There are four sides to the box each with an area of xy sq. feet. So the cost of one side is 4xy. But there are 4 sides, so total cost of the sides is 16xy.

 

So we have a cost function that looks like this: C = 10x² + 16xy. We need to find the derivative of C with respect to one of the independent variables. That is why we were given the information about the volume of the box, in order to establish a relationship between x and y. Since the volume is 20 cubic feet we have 20 = x²y (lwh). That means y = 20x^-2.

 

So now the cost function can be written as:

 

C= 10x² + 16x(20x^-2) = 10x² + 320x^-1.

 

Find dC/dx, set the derivative equal to 0 to and solve for x. The values for x represent the critical numbers of the cost function. According to the way the problem is worded apparently that value of x will produce a minimum cost for the box.