Step by step
The question
Question: two similar solids have matching lengths 2 and 4, and the small volume is 10: find the large volume.
lengths 2 and 4, small volume = 10
1 Length scale factor
The scale factor k: every length of the large one is 2 times the small one.
k = 42 = 2
2 Volume factor
Volume grows in three directions: k times k times k.
That is 8.
volume factor = k³ = 2³ = 8
A cube 1 by 1 by 1, made 2 times longer every way, holds 8 little cubes.
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 10 × 2 ✗
3 Use it
10 times 8 is 80.
large volume = 10 × 8 = 80
4 Check
Let's check: the large volume divided by the small one is 8, which is k cubed.
So it is right!
80 ÷ 10 = 8 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
large volume = 10 × 8 = 80