Question

A closed box with a square base is to have a volume of 16,000 cm^3. The material for the top and bottom of the box costs $3 per square centimeter, while the material for the sides costs $1.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost ?

Answer 1

Answer:

base side = 20cm

height = 40 cm

minimum cost = $7200

Step-by-step explanation:

Let x (cm) be the base side and h (cm) be the height of the box. Since the volume of the box must be 16000 cubic cm, we have:

$x^2h = 16000$

$h = 16000/x^2$

The areas of the top and bottom is

$A_B = 2x^2$

And so their cost is:

$C_B = A_T*3 = 6x^2$

The areas of the sides is

$A_S = 4xh$

And so their cost is:

$C_S = A_S*1.5 = 6xh$

We can substitute $h = 16000/x^2$ into the equation above to get

$C_S = 96000/x$

So the total cost of the bases and the sides:

$C = C_S + C_B = 6x^2 + 96000/x$

This is the function we would want to minimize. We can start by taking the 1st derivative, and set it to 0

$C^{'} = 12x - 96000/x^2 = 0$

$12x = 96000/x^2$

$x^3 = 96000/12 = 8000$

$x = 20cm$

So the height would be

$h = 16000/x^2 = 16000/20^2 = 40 cm$

So the minimum total cost would be

$C = 6x^2 + 96000/x = 6*20^2 + 96000/20 = 7200$