Question

A container manufacturer plans to make rectangular boxes whose bottom and top measure 4x by 3x. The container must contain 48in.3 The top and the bottom will cost $1.80 per square inch, while the four sides will cost $3.70 per square inch.
What should the height of the container be so as to minimize cost? Round your answer to the nearest hundredth.

Answer 1

The height of the container that will be able to minimize the cost will be 3.08cm.

How to calculate the height?

The volume of the box will be:

= (3x)(4x)h

= 12x²h

From the information given, we are told that the container must contain 48in³. Therefore,

48 = 12x²h

h = 4/x²

The function cost will be:

= 3.50(2)(12x²) + 4.40(14x)h

= 84x² + 61.6x(4/x²)

= 84x² + 246.4/x

We'll use the first derivative. This will be:

dC/dx = 168x - 246.4/x²

x = 1.14.

Therefore, the height will be:

h = 4/x² = 4/1.14² = 3.08cm

In conclusion, the height is 3.08cm.

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