A box with a square base and open top must have a volume of 32,000cm³. How do you find the dimensions of the box that minimize the amount of material used?

Solution:

It is given that

Volume of box = 32,000 cm³

Consider b as the square base and h as the height of the box

The formula to find the volume is

V = b²h

h = V/b²

The formula to find the surface area is

A = b² + 4b x V/b²

A = b² + 4V/b

By differentiating A with respect to b to find the maxima or minima

dA/db = 2b - 4V/b²

2b - 4V/b² = 0

b³ = 4V/2

Substituting the values

b³ = 4 (32000)/2

So we get,

b = 40 cm

h = 32000/ 402 = 20 cm

Therefore, the dimensions of the box are 40 cm and 20 cm.

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A box with a square base and open top must have a volume of 32,000cm³. How do you find the dimensions of the box that minimize the amount of material used?

Summary:

A box with a square base and open top must have a volume of 32,000cm³. The dimensions of the box that minimize the amount of material used are 40 cm and 20 cm.