An engineer must design a rectangular box that has a volume of 9 m3 and that has a bottom whose length is twice its width. What are the dimensions of the box so that the total surface area (of all six sides) of the box is minimized

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Explanation:

Given

Represent the height as h, the length as l and the width as w.

From the question:

Volume of a box is calculated as:

This gives:

Substitute 9 for V

Make h the subject:

The surface area is calculated as:

Recall that:

Recall that:

So:

To minimize the surface area, we have to differentiate with respect to w

Set A' to 0

Add to both sides

Multiply both sides by

Make the subject

Solve for w

Recall that :   and

Hence, the dimension that minimizes the surface area is:

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