Step by step
The question
Question: two similar solids have matching lengths 2 and 3, and the small surface area is 20: find the large surface area.
lengths 2 and 3, small surface area = 20
1 Length scale factor
The scale factor k: every length of the large one is 3/2 times the small one.
k = 32
2 Area factor
Area grows in two directions: k times k.
That is 9/4.
area factor = k² = (32)² = 94
Careful!
The surface area does not grow like the lengths.
It grows by k squared.
Careful: not 20 × 3/2 ✗
3 Use it
20 times 9/4 is 45.
large surface area = 20 × 94 = 45
4 Check
Let's check: the large surface area divided by the small one is 9/4, which is k squared.
So it is right!
45 ÷ 20 = 9/4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
large surface area = 20 × 94 = 45