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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

  • Unsure what you are asking. Since (for example) there are $(100)$ centimeters in a meter, there are $(100)^3 = 1,000,000$ cubic centimeters in a cubic meter. Does this help? – user2661923 Aug 22 '21 at 23:47
  • I am unfamiliar with the abbreviation dm. Does this refer to $10$ meters? If so, then re previous comment, there are $(1000)$ centimeters in a dm and therefore there are $(1000)^3$ centimeters in a cubic dm. What is a dm and what is the relationship between a cubic dm (or a cubic meter, for that matter) and a liter? – user2661923 Aug 22 '21 at 23:51
  • Alternatively, if dm refers to $0.1$ meters, then there are $10$ cm in a dm and thus there are $(10)^3$ cubic centimeters in a cubic dm. – user2661923 Aug 22 '21 at 23:53
  • Based on the comment of @CyclotomicField you can ignore my 3rd comment and focus on my second comment. – user2661923 Aug 22 '21 at 23:55
  • $$1~\operatorname{dm} = 10~\operatorname{cm} \implies 1~L = 1~\operatorname{dm}^3 = 1000~\operatorname{cm}^3 \implies 28080~L = 1000 \times 28080~\operatorname{cm}^3 = 28080000~\operatorname{cm}^3.$$ Can you now find the scale? – an4s Aug 22 '21 at 23:56
  • Another hint: when the linear dimension is multiplied by10, the volume is multiplied by $10^3=1000$. The scale is the linear reduction factor. – Jean-Claude Arbaut Aug 23 '21 at 00:01
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    @CyclotomicField It's the other way around: 1 decimeter = 0.1 meter, and 1 decameter = 10 meters. – Jean-Claude Arbaut Aug 23 '21 at 00:05
  • Guys, thanks for the help. Apparently the problem is that I had to convert dm³ to cm, so 1000dm³ becomes 100cm. I believe that's the idea, anyway, the correct answer is alternative B, however, I'm not sure my calculation is right. Thank you all for your support. –  Aug 23 '21 at 00:19
  • @Jean-ClaudeArbaut you're right I'll fix it. – CyclotomicField Aug 23 '21 at 00:26

1 Answers1

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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.

Jean-Claude Arbaut
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