A closed box with a square base is to have a volume of 85. 75 cm^3. the material for the top and bottom of the box costs $3.00 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?

We can create two equations here:

(1) Volume = area of square * height of box

85.75 = s^2 h

(2) Cost = 3 * area of square + 1.5 * area of side box

C = 3 s^2 + 1.5 s h

From (1), we get:

h = 85.75 / s^2

Combining this with (2):

C = 3 s^2 + 1.5 s (85.75 / s^2)

C = 3 s^2 + 128.625 s-

Taking the 1st derivative and equating dC/ds = 0:

dC/ds = 6s - 128.625 / s^2 = 0

Multiply all by s^2:

6s^3 - 128.625 = 0

6s^3 = 128.625

s = 2.78 cm

So h is:

h = 85.75 / s^2 = 85.75 / (2.78) ^2

h = 11.10 cm

So the dimensions are 2.78 cm x 2.78 cm x 11.10 cm

The total cost now is:

C = 3 (2.78) ^2 + 1.5 (2.78) (11.10)

C = $69.47