Question

How tall is a stack of three cube-shaped boxes, one on top of the other, given that their volumes are 24 cubic inches, 81 cubic inches and 375 cubic inches? Give an exact answer.

Answer 1

The volume of a cube with side length x is given by
`x^3`.

So, if the volume of a cube is
`V`, we find its side length by taking the cube root of V.

Thus, the side lengths of the cubes are:


`\sqrt[3]{24}= \sqrt[3]{8\cdot3}= \sqrt[3]{8}\cdot\sqrt[3]{3}=2\sqrt[3]{3}`


`\sqrt[3]{81}= \sqrt[3]{27\cdot3}= \sqrt[3]{27}\cdot\sqrt[3]{3}=3\sqrt[3]{3}`


`\sqrt[3]{375}= \sqrt[3]{125\cdot3}= \sqrt[3]{125}\cdot\sqrt[3]{3}=5\sqrt[3]{3}`

Thus, the stack has a height of
`2\sqrt[3]{3}+3\sqrt[3]{3}+5\sqrt[3]{3}=10\sqrt[3]{3}`.


Answer:
`10\sqrt[3]{3}`