Step by step
The question
Question: two similar solids have matching lengths 2 and 5, and the small volume is 16: find the large volume.
lengths 2 and 5, small volume = 16
1 Length scale factor
The scale factor k: every length of the large one is 5/2 times the small one.
k = 52
2 Volume factor
Volume grows in three directions: k times k times k.
That is 125/8.
volume factor = k³ = (52)³ = 1258
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 16 × 5/2 ✗
3 Use it
16 times 125/8 is 250.
large volume = 16 × 1258 = 250
4 Check
Let's check: the large volume divided by the small one is 125/8, which is k cubed.
So it is right!
250 ÷ 16 = 125/8 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
large volume = 16 × 1258 = 250