Video Transcript
Consider a cube that initially has
side length 𝑥. If 𝑥 increases by a factor of two,
by what factor does the volume of the cube change? By what factor does the surface
area of the cube change?
Alright, so here’s our cube that
we’re told initially has a side length 𝑥. And then, we’re to imagine that 𝑥
increases by a factor of two. This would lead to a cube that
looks like this, with side length two 𝑥. When this change happens, we first
want to know, by what factor does the volume of the cube change? Now, if we have a cube where 𝑙 is
the length of each side, then we can recall that the volume of that cube is equal to
𝑙 cubed. This means that if we call the
volume of the cube with side length 𝑥 “𝑉 sub 𝑥”, then by our equation for the
volume of a cube, that volume is equal to 𝑥 cubed, while the volume of the cube
with side length two 𝑥, we can call it 𝑉 sub two 𝑥, is equal to two 𝑥 quantity
cubed or eight times 𝑥 cubed.
To answer our question, to say by
what factor the volume of the cube changes, we’ll want to divide our final cube
volume by our initial volume. When we do this, we see that it’s
equal to eight times 𝑥 cubed divided by 𝑥 cubed. So, the factors of 𝑥 cubed cancel,
telling us that when we double the side length of our cube from 𝑥 to two 𝑥, its
volume has increased by a factor of eight. This is the answer to the first
part of our question.
In part two, we want to say by what
factor does the surface area of the cube change. If we again consider a cube of side
length capital 𝐿, then the surface area of that cube is equal to six times 𝐿
squared. This is because each of the cube’s
faces has an area of 𝐿 squared and there are six faces in total. So, if we call 𝐴 sub 𝑥 the
surface area of our cube when it’s side length is 𝑥, then that’s equal to six times
𝑥 squared. And then, if 𝐴 sub two 𝑥 is the
cube’s surface area after its side length has doubled, then that’s equal to six
times two 𝑥 quantity squared or six times four times 𝑥 squared, which equals 24𝑥
squared.
Once more, we’ll want to take the
ratio of the final quantity to the initial quantity. When we divide 24𝑥 squared by six
𝑥 squared, the 𝑥 squareds cancel out. And then, 24 divided by six is
four. So, this is our answer to how the
surface area of the cube changes with this change in side length. All together then, when we double
the side length of a cube, its volume increases by a factor of eight and its surface
area grows by a factor of four.
