Step by step
The question
Question: two similar solids have volumes 1 and 8, and the small radius is 2.5: find the large radius.
volumes 1 and 8, small radius = 2.5
1 Volume factor
The large volume divided by the small one: 8.
That is k cubed.
volume factor = 81
2 Length scale factor
To go back from k cubed to k, we take the cube root: 2.
k = ∛8 = 2
2 × 2 × 2 = 8
Careful!
8 is the volume factor.
Lengths grow less: by its cube root.
Careful: not k = 8 ✗
3 Use it
2.5 times 2 is 5.
large radius = 2.5 × 2 = 5
4 Check
Let's check: the large radius divided by the small one is 2, the scale factor.
So it is right!
5 ÷ 2.5 = 2 = k
Let's look back.
First: volume factor.
Then: length scale factor.
Then: use it.
Last: check.
Answer
large radius = 2.5 × 2 = 5