3D ratio math help needed.

Question

There are three cuboids, A, B and C, the length of A is (1/2) the length of B, and (1/3) the length of C, the width of of A is (2/3) the width of B and (3/2) the width of C and the height of A is the same as the height of B and 3 times the height of C. Find the ratio of the surface area of the cuboids A, B and C.

Answer 1

Let 

L = length of cuboid A, W = width of cuboid A, H = height of cuboid A

L = (1/2)L_b = (1/3) L_c 

W = (2/3)W_b = (3/2) W_c

H = H_b = 3H_c

The dimensions of cuboids A, B, and C in terms of L, W and H are as follows:

(1) Cuboid A: L, W and H

(2) Cubiod B: 2L, (3/2)W and H

(3) Cuboid C: 3L, (2/3)W and (1/3)H

First, ratio for the volume:

V_a = LWH

V_b=2L•(3/2)W•H =3LWH

V_c=3L•(2/3)W•(1/3)H =(2/3)LWH

The ratio of the Volume is 

A:B:C=1:3:2/3 = 3:9:2

Let’s think it this way, 2X2X2 cube and 3X3X3 cube has a side ratio of 2:3, a surface area ratio of 4:9 and volume ratio of 8:27. The pattern is square the side ratio to get the surface area ratio, then cube the side ratio to get the volume ratio. Therefore if you want to get the ratio of the surface area from the volume ratio, simply raise every number by 2/3 (same as get the cube root and then square it)

SA_a:SA_b:SA_c

3^2/3:9^2/3:2^2/3

=9^1/3:81^1/3:4^1/3

Answer 2

I don't think this question can be answered as written. Here is a quick explanation why:

I'm going to use sub L, W, and H to indicate length, width, and height respectively for each cuboid A, B, C

A_L = 1/2B_L = 1/3C_L

A_W = 2/3B_W = 3/2C_W

A_H = B_H = 3C_H

We can then solve these to get values for the other cuboid's sides as a function of A's corresponding sides.

B_L = 2A_L

B_W = 3/2A_W

B_H = A_H

C_L = 3A_L

C_W = 2/3A_W

C_H = 1/3A_H

If we assume that A is a cube with all sides equal to 1, we can then plug that value in to find the length of all sides and then solve the surface area equation for cuboids to find their areas:

Surface area equation: SA = 2(lw + lh + hw)

Solving for surface area here, we get:

Surface area A = 6

Surface area B = 13

Surface area C = 6⁴/₉

Not exactly pretty, but this could give us some sort of system of ratios. The problem arises from the fact that we are dealing with cuboids specifically, and they have no addition constraints to their growth per the description of the problem, meaning that all we have to do is sub in 2 for A_L and suddenly the surface areas change like this:

Surface area A = 10

Surface area B = 23

Surface area C = 12⁴/₉

As you can tell 6:13 ≠ 10:13. As a result, this problem has no clear answer without either:

a.) a constraint on the dimensions of the cuboids relative to their other sides.

b.) values to plug in for each side.