optimization question

Question

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.29 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.21 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

Answer 1

Write the equations for volume and surface area.  The surface area will be used to create a function for cost.

 

Let length = width = x

Let height = y

 

Using these variables to create the equations for volume and cost.

 

x²y = 40     ------->  volume

 

C = 0.29x² + 0.21x² + 0.05(4xy)      -----> cost

 

C = 0.50x² + 0.05xy         ------> cost simplified

 

 

Substitute the volume equation into the cost function.  The cost function will only be in terms of x.

 

C = 0.5x² + 0.05x(40 / x²)

 

C = 0.5x² + 2x^-1

 

 

Now, take the derivative of C and set it equal to zero.  Recall the first derivative test.

 

d/dx[C] = 0

 

 

Solve for x from the derivative equation.  The x values you get will be your critical points.  Then you will need to use test points to see which critical point has a minimum.  That means the derivative before the critical point is negative, and the derivative after the critical point is positive.