Step by step
The question
Question: two similar solids have volumes 27 and 64, and the small height is 6: find the large height.
volumes 27 and 64, small height = 6
1 Volume factor
The large volume divided by the small one: 64/27.
That is k cubed.
volume factor = 6427
2 Length scale factor
To go back from k cubed to k, we take the cube root: 4/3.
k = ∛(6427) = 43
4/3 × 4/3 × 4/3 = 64/27
Careful!
64/27 is the volume factor.
Lengths grow less: by its cube root.
Careful: not k = 64/27 ✗
3 Use it
6 times 4/3 is 8.
large height = 6 × 43 = 8
4 Check
Let's check: the large height divided by the small one is 4/3, the scale factor.
So it is right!
8 ÷ 6 = 4/3 = k
Let's look back.
First: volume factor.
Then: length scale factor.
Then: use it.
Last: check.
Answer
large height = 6 × 43 = 8