Question

A container manufacturer plans to make rectangular boxes whose bottom and top measure x by 4x. The container must contain 12ft^3. The top and the bottom will cost $1.50 per square foot, while the four sides will cost $4.50 per square foot.
What should the height of the container be so as to minimize cost? Round your answer to the nearest hundredth. The answer should be written in ft.

Answer 1

The height of the container should be equal to 2.57 cm, so as to minimize cost.

How to calculate the height?

First of all, we would determine the volume of the rectangular container by using this formula:

Volume = l × w × h

Where:

• l represents the length.

• w represents the width.

• h represents the height.

Substituting the given parameters into the formula, we have;

Volume = x × 4x × h

Volume = 4x²h

At a volume of 12 ft³, we have:

12 = 4x²h

h = 12/4x²

h = 3/x²

For the total area of top and bottom, we have:

Area = (x × 4x) + (x × 4x)

Area = 4x² + 4x²

Area = 8x² sq. units.

Thus, the cost of top and bottom is given by:

Cost = 8x² × 1.50

Cost = $12x²

For the total area of all four sides, we have:

Area = (x × h) + (x × h) + (4x × h) + (4x × h)

Area = xh + xh + 4xh + 4xh

Area = 10xh sq. units.

Thus, the cost of all four sides is given by:

Cost = 10xh × 4.50

Cost = $45xh.

Also, the cost function would be given by:

C(x) = 12x² + 10xh

C(x) = 12x² + 10x(3/x²)

C(x) = 12x² + 30/x

Taking the first derivative, we have:

dC/dx = 24x - 30/x²

24x = 30/x²

24x³ = 30

x³ = 30/24

x = 1.08

Therefore, the height will be:

h = 3/x²

h = 3/1.08²

h = 3/1.1664

Height, h = 2.57 cm.

Read more on minimum height here: brainly.com/question/27308167

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