Step by step
The question
Question: two similar solids have matching lengths 1 and 3, and the small volume is 5: find the large volume.
lengths 1 and 3, small volume = 5
1 Length scale factor
The scale factor k: every length of the large one is 3 times the small one.
k = 31 = 3
2 Volume factor
Volume grows in three directions: k times k times k.
That is 27.
volume factor = k³ = 3³ = 27
A cube 1 by 1 by 1, made 3 times longer every way, holds 27 little cubes.
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 5 × 3 ✗
3 Use it
5 times 27 is 135.
large volume = 5 × 27 = 135
4 Check
Let's check: the large volume divided by the small one is 27, which is k cubed.
So it is right!
135 ÷ 5 = 27 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
large volume = 5 × 27 = 135