Step by step
The question
Question: two similar solids have matching lengths 4 and 6, and the large volume is 135: find the small volume.
lengths 4 and 6, large volume = 135
1 Length scale factor
The scale factor k: every length of the large one is 3/2 times the small one.
k = 64 = 32
2 Volume factor
Volume grows in three directions: k times k times k.
That is 27/8.
volume factor = k³ = (32)³ = 278
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 135 × 3/2 ✗
3 Use it
Going to the small one, we divide: 135 divided by 27/8 is 40.
small volume = 135 ÷ 278 = 40
4 Check
Let's check: the large volume divided by the small one is 27/8, which is k cubed.
So it is right!
135 ÷ 40 = 27/8 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
small volume = 135 ÷ 278 = 40