Step by step
The question
Question: two similar solids have volumes 8 and 27, and the large height is 9: find the small height.
volumes 8 and 27, large height = 9
1 Volume factor
The large volume divided by the small one: 27/8.
That is k cubed.
volume factor = 278
2 Length scale factor
To go back from k cubed to k, we take the cube root: 3/2.
k = ∛(278) = 32
3/2 × 3/2 × 3/2 = 27/8
Careful!
27/8 is the volume factor.
Lengths grow less: by its cube root.
Careful: not k = 27/8 ✗
3 Use it
9 divided by 3/2 is 6.
small height = 9 ÷ 32 = 6
4 Check
Let's check: the large height divided by the small one is 3/2, the scale factor.
So it is right!
9 ÷ 6 = 3/2 = k
Let's look back.
First: volume factor.
Then: length scale factor.
Then: use it.
Last: check.
Answer
small height = 9 ÷ 32 = 6