I'm having trouble solving a problem involving volumetric scale I'm stuck on an issue involving volumetric scale. I tried to multiply the dimensions 2 cm × 3.51 cm × 4 cm, I got the result 28.08, and I believe the result must contain cm³, because I multiplied the factors 3 times... The problem is that I can't find the scale.
I tried to do the following:
2808 / 28080
I divided 2808 by itself, getting the result 1. Then, dividing 28080 by 2808, I got 1000 as a result, so the scale I found was 1:1000, but this is not the correct answer. I think my reasoning makes sense, however, I can't get to the answer. I hope someone can help me and guide me about my mistakes
Question:
The water tank of a building will have the shape of a
straight rectangle parallelepiped with volume equal to
28 080 liters. In a model that represents the building,
the water tank has dimensions 2 cm × 3.51 cm × 4 cm.
Given: 1 dm³ = 1 L.
The scale used by the architect was
a) 1 : 10 
b) 1 : 100 
c) 1 : 1 000 
d) 1 : 10 000 
e) 1 : 100 000 
 A: From your last comment, even though you have the correct answer, you don't know why, and worst of all, you are mixing volumic and linear units.
Let's try to do it correctly.
For linear units: $1\,\mathrm{m} = 10\,\mathrm{dm} = 100\,\mathrm{cm} = 1000\,\mathrm{mm}$.
In two dimensions, the factors get a "square" exponent (hence the name - square):
For area units: $1\,\mathrm{m}^2 = 10^2\,\mathrm{dm}^2 = 100^2\,\mathrm{cm}^2 = 1000^2\,\mathrm{mm}^2$.
That's because, for instance, a square with a $10\,\mathrm{dm}$ side can be cut in a $10\times10$ grid of $100$ smaller squares of side $1\,\mathrm{dm}$.
Or more simply, $1\,\mathrm{m}^2=10\,\mathrm{dm}\times10\,\mathrm{dm}=10\times10\times\mathrm{dm}\times\mathrm{dm}=100\,\mathrm{dm}^2$.
Likewise, in three dimensions, the factor gets a "cube" exponent (hence the name, cube):
For volume units: $1\,\mathrm{m}^3 = 10^3\,\mathrm{dm}^3 = 100^3\,\mathrm{cm}^3 = 1000^3\,\mathrm{mm}^3$.
And the $\mathrm{dm}^3$ unit has a friendly name, it's a liter. So a liter is $1000\,\mathrm{cm}^3$. It's equivalent to a cube of side $10\,\mathrm{cm}$.
Now, you have a volume of 28,080 liters, hence 28,080,000 $\mathrm{cm}^3$, and a model with a volume $28.08\,\mathrm{cm}^3$. That is, the volume of the model is 1,000,000 times smaller.
Given a rectangular parallelepiped with sides $a,b,c$ (hence volume $abc$), if you multiply the sides by $10$, the volume is $10a\times10b\times10c=1000abc$, hence the volume is multiplied by $10^3=1000$.
Therefore, to multiply the volume by 1,000,000, you need to multiply the sides by 100. That is, the model has linear dimensions (i.e. sides) 100 times smaller than the real water tank.
That is, the scale is 1:100.
