Question

A box with a square base and open top must have a volume of 32,000cm^3. How do you find the dimensions of the box that minimize the amount of material used?

Answer 1

Answer:

Side of 40 and height of 20

Step-by-step explanation:

Let s be the side of the square base and h be the height of the box. Since the box volume is restricted to 32000 cubic centimeters  we have the following equation:

$V = hs^2 = 32000$

$h = 32000/ s^2$

Assume that we cannot change the thickness, we can minimize the weight by minimizing the surface area of the tank

Base area with open top $s^2$

Side area 4hs

Total surface area $A = s^2 + 4hs$

We can substitute $h = 32000/ s^2$

$A = s^2 + 4s\frac{32000}{s^2}$

$A = s^2 + 128000/s$

To find the minimum of this function, we can take the first derivative, and set it to 0

$A' = 2s - 128000/s^2 = 0$

$2s = 128000/s^2$

$s^3 = 64000$

$s = \sqrt[3]{64000} = 40$

$h = 32000/ s^2 = 32000/ 40^2 = 20$