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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