7. The volume of a cube in cubic metres and its surface area in square metres is numerically equal to four-thirds of the sum of the lengths of its edges in metres.
What is the total volume in cubic metres of twenty-seven such cubes?
Solution
216

Let each of the twelve edges of the cube have length $x$ metres. Then $x^{3}+6 x^{2}=\frac{4}{3} \times 12 x$. This simplifies to $x^{3}+6 x^{2}-16 x=0$ or $x(x-2)(x+8)$ which has solutions $x=-8,0,2$. However $x$ must be positive and so $x=2$. Then 27 such cubes have a volume of $27 \times 2^{3}=27 \times 8=216$.