Question

A cereal box, in the shape of a rectangular prism and with a closed top, is to be
constructed so that the base is twice as long as it is wide. Its volume is to be 8000cm3。

Find the dimensions that will minimize the amount of cardboard required to make the box.

Answer 1

Solution: Given the following rectangular prism:

Since the base is twice as long as it is wide, therefore,

... (1)

The volume of the cuboid is given by:

Using equation (1), we'll get:

Since the volume given is 8000 , therefore,

... (2)

The objective is to minimize the amount of cardboard required to make the box, therefore, we've to minimize the surface area S in order to reduce the amount of cardboard to be used. The surface area is given by:

Using equation (1) and (2), we'll get:

Now, we'll differentiate this equation w.r.t B to find the critical points of the function S.

Equating this equation to zero:

Substituting in equation (1) and (2), we'll get:

and

Thus, we got the required dimensions 28.84 cm x 14.42 cm x 19.93 cm.

For the function S(B), we haven't used the second derivative test to verify whether the B = 14.42 cm will give the minimum value of the function because we just got a single critical point. If we get two points then we've to use the second derivative to check which one is the point of minimum.

I hope it helps you!