Question:
If the length of each edge of a cube is doubled, how many times does its volume become? How many times does its surface area become?
Solution:
Let $a$ be the length of the edge of a cube.
Volume of the cube $=a^{3}$
Total surface area $=6 a^{2}$
If the length is doubled, then the new length becomes $2 a$.
Now, new volume $=(2 a)^{3}=8 a^{3}$
Also, new surface area $=6(2 a)^{2}=6 \times 4 a^{2}=24 a^{2}$
$\therefore$ The volume is increased by a factor of 8 , while the surface area increases by a factor of 4 .