Question

A rectangular box without a lid should be made with 12m2 of cardboard. What are the dimensions of the box that maximize the volume?

a.) 2m x 2m x 2m

b) 1.54m x 1.54m x 0.77m

c) 2m x 2m x 1m

d) 4m x 4m x 2m

Answer 1

Solution: A rectangular box without a lid is to be made from the cardboard of area 12 , therefore, the surface area of the rectangular box with dimension l x b x h must be equal to the area of the cardboard.

... (1)

We haven't multiplied the term l.b by 2 because the box doesn't have a lid.

The objective is to find the dimensions of the box that maximizes the volume. Therefore, we've to maximize the equation below subject to the equation (1),

We'll use Lagrange's Multiplier to maximize the volume subject to the equation (1). To do so, we've to solve the following equations simultaneously:

The gradient of the function is given by:

Therefore,

Therefore,

Comparing each component:

... (2)

... (3)

... (4)

Dividing equation (2) and (3), we'll get:

Substituting this value in equation (4), we'll get:

Since l can't be zero, therefore,

Substituting this value in equation (3), we'll get:

Therefore, we've:

Substituting this value in equation (2) and simplifying, we'll get:

We'll neglect the negative value because x, y, z > 0. Therefore,

Therefore,

Thus, option (c) is the correct answer.

I hope it helps you!