Step by step
The question
Question: two similar solids have matching lengths 3 and 6, and the small volume is 20: find the large volume.
lengths 3 and 6, small volume = 20
1 Length scale factor
Similar means the same shape: one is a bigger copy.
The scale factor k says how many times longer: the large one over the small one, 6/3, that is 6 divided by 3: 2.
k = 63 = 2
2 Volume factor
Volume grows in three directions: k times k times k.
2 times 2 times 2 is 8.
volume factor = k³ = 2³ = 8
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 20 × 2 ✗
A cube 1 by 1 by 1, made 2 times longer every way, holds 8 little cubes.
3 Use it
Going to the large one, we multiply by the factor: 20 times 8 is 160.
large volume = 20 × 8 = 160
4 Check
Let's check: the large volume divided by the small one is 8, which is k cubed.
So it is right!
160 ÷ 20 = 8 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
large volume = 20 × 8 = 160