Proportionality Question: ""Freight transport on Titan is mostly by ship...."

[S_100]

"Freight transport on Titan is mostly by ship, with three types of ship called pangs, quizzers, and roodles in common use. All three ships have the same shape and design but differ in size. The cargo capacity depends on the hold volume, while the number of crew required is proportional to the surface area of the deck.

A quizzer and a roodle taken together have the same length as two pangs, and the crew of a quizzer is just sufficient to provide crew for two pangs and a roodle.

A fully loaded quizzer wishes to transfer all its cargo to smaller pangs and roodles, while minimising the number of crew required for the resultant fleet.
How many pangs and roodles are needed?

[Hint: Note that for objects of any shape the surface area is proportional to the square of the object’s size, and the volume is proportional to the cube of its size.] "


My Solution

no.of crew ∝ surface area -------------> surface area∝size^2∝ L^2 where L is length of ship no.of crew = C
therefore,
C ∝ L^2

cargo capacity ∝ volume of hold ∝ size ∝ L^3
so

cargo capacity∝L^3

then ,

let (L) length of Ship be denoted by Intial letter of type of ship, Pang, Quizzer, Roodle

Q + R = 2P

C_Q = 2 C_P + C_R

Knowing C ∝ L^2

then C_Q = 2 C_P + C_R ---> Equivalent to -----> Q² = 2(P² + R²

This is the part I'm confused on, My Question:
why is

2C_P ∝ 2P² instead of ∝ 4P²

I always end up making this mathematical error, so I would like to know why the latter case (4P^2) is wrong mathematically, also even if I didn't understand it mathematically, is there an intuitive way to realise the 4P^2 is wrong from the question

 

[Dr.Peterson]

2C_P ∝ 2P² and 2C_P ∝ 4P² are both true!

I think the problem may be that you are forgetting the constant of proportionality.

But to make sure I'm understanding your thinking, can you tell us why you think it should be 4P²? And can you show us the entire work as you think it would be done?