Step by step
The question
Question: two similar solids have matching lengths 3 and 4, and the small volume is 54: find the large volume.
lengths 3 and 4, small volume = 54
1 Length scale factor
The scale factor k: every length of the large one is 4/3 times the small one.
k = 43
2 Volume factor
Volume grows in three directions: k times k times k.
That is 64/27.
volume factor = k³ = (43)³ = 6427
Careful!
The volume does not grow like the lengths.
It grows by k cubed.
Careful: not 54 × 4/3 ✗
3 Use it
54 times 64/27 is 128.
large volume = 54 × 6427 = 128
4 Check
Let's check: the large volume divided by the small one is 64/27, which is k cubed.
So it is right!
128 ÷ 54 = 64/27 = k³
Let's look back.
First: length scale factor.
Then: volume factor.
Then: use it.
Last: check.
Answer
large volume = 54 × 6427 = 128