A rectangular storage container with an open top is to have a volume of 10 m^3

Question

A rectangular storage container with an open top is to have a volume of 10 m^3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

Answer 1

The area of each face of the container has a certain cost.  The volume is base time width times height.  Because we have an open top, there will be 1 base area, and 4 side areas.  2 of the side areas are equal to each other, and the other 2 side areas are also equal to each other.

 

Let base length = 2x
Let width = x
Let height = h

 

Using these variables, we set an equation for volume and cost.

 

10 = 2x²h  -----> volume

 

C = 10(2x²) + 6[2*2xh + 2xh]

 

C = 20x² + 6(4xh + 2xh)

 

C = 20x² + 6(6xh)

 

C = 20x + 36xh    ------> cost

 

 

Substitute the volume equation into the cost equation so that the cost equation is in terms of x.

 

C = 20x + 36x(5 / x²)

 

C = 20x + 80x^-1

 

 

Now, we can take the derivative of C and set it equal to zero.  This is how we find the cheapest cost because we want to find the minimum cost.

 

20 - 80x^-2 = 0

 

20 - (80 / x²) = 0

 

(20x² - 80) / x² = 0

 

 

Set the numerator equal to zero.

 

20x² - 80 = 0

 

20x² = 80

 

    x² = 4

 

    x = 2

 

The cost of materials will be cheapest when the width of the container is 2 meters, base length is 4 meters, and the height is 1.25 meters.

 

Plug in these values into the cost equation to find the total cost:

 

x = 2

h = 1.25