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Differentiation: Stationary Points Example 3

Question:

A box without a lid has a square base of side 𝑥 and rectangular sides of height ℎ . It is made of 10800mm2 of sheet metal of negligible thickness. Prove that:

and that the volume of the box is:

Hence calculate the maximum volume of the box.


Answer:

We know the standard formula to calculate the volume of a rectangular box:

Now differentiate and set equal to zero to find the turning points:

Plug the value for x into the second derivative to find out if this x value will produce a maximum value for the volume of the box.


Now simply sub x into original equation for the max volume:


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