Question

How much metal is needed to cast a cubical metal box with outer side $10\mathrm{inches}$ long if the thickness of its walls should be exactly $2\mathrm{inches}$?

Answer 1

Finding the volume of the solid box.
The formula for the volume of a cube is ${\left(\mathrm{side}\right)}^3$.

Here the side of the cube is $10\mathrm{inches}$.

$\begin{array}{rcl}\mathrm{V}& =& {10}^3\\ \therefore \mathrm{V}& =& 1000\end{array}$
If the side of the cube is $10\mathrm{inches}$, and the width of the walls is $2\mathrm{inches}$, then the width of the cube of air inside of the box is $10-2\left(2\right)=6\mathrm{inches}$.
The volume of that box is $6^3=216$
Thus, the amount of metal to cast the box is $1000-216=784{\mathrm{inches}}^3$.

Hence, the amount of metal needed is $784{\mathrm{inches}}^3$.