Step by step
The question
Question: two similar solids have matching lengths 3 and 6, and the large volume is 80: find the small volume.
lengths 3 and 6, large volume = 80
1 Length scale factor
The scale factor k: every length of the large one is 2 times the small one.
k = 63 = 2
2 Volume factor
Volume grows in three directions: k times k times k.
That is 8.
volume factor = k³ = 2³ = 8
A cube 1 by 1 by 1, made 2 times longer every way, holds 8 little cubes.
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 80 × 2 ✗
3 Use it
Going to the small one, we divide: 80 divided by 8 is 10.
small volume = 80 ÷ 8 = 10
4 Check
Let's check: the large volume divided by the small one is 8, which is k cubed.
So it is right!
80 ÷ 10 = 8 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
small volume = 80 ÷ 8 = 10