A closed box with a square base is to have a volume of 171 comma 500 cm cubed. The material for the top and bottom of the box costs $10.00 per square centimeter, while the material for the sides costs $2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost? Write an equation for C(x), the cost of the box as a function of x, the length of a side of the base. C(x)equals nothing

The dimensions that will lead to minimum cost of the box are a base length of 35 cm and a height of 140 cm.

Step-by-step explanation:

Volume of the Square-Based box=171,500 cubic cm

Let the length of a side of the base=x cm

Volume

The material for the top and bottom of the box costs $10.00 per square centimeter.

Surface Area of the Top and Bottom

Therefore, Cost  of the Top and Bottom

The material for the sides costs $2.50 per square centimeter.

Surface Area of the Sides=4xh

Cost of the sides=$2.50 X 4xh =10xh

Therefore, total Cost of the box

To find the minimum total cost, we solve for the critical points of C(x). This is obtained by equating its derivative to zero and solving for x.

Recall that:

The dimensions that will lead to minimum costs are base length of 35cm and height of 140cm.

Therefore, the minimum total cost, at x=35cm

step-by-step explanation: