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navik [9.2K]
4 years ago
5

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?
Mathematics
1 answer:
sveticcg [70]4 years ago
8 0

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

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