Which Object is the Most Dense Based on Given Measurements?

[Drakkith]

Homework Statement

Of the objects below, which is the most dense?

A. an object with a volume of 13 dm³ and a mass of 1.29 × 10³ g
B. an object with a volume of 0.00212 m³ and a mass of 4.22 × 10⁴ mg
C. an object with a volume of 3.91 × 10^-24 nm³ and a mass of 7.93 × 10^-1 ng
D. an object with a volume of 139 mL and a mass of 93 g
E. an object with a volume of 2.5 L and a mass of 12.5 kg

Homework Equations

Density = mass/volume

The Attempt at a Solution

I'm having trouble converting my units of volume so I can calculate these, especially with C. How do you get from cubic nanometers to cubic centimeters? I know it's not as easy as just dividing by 10⁷. I'm guessing you need to find out how many nanometers are in a centimeter and then cube the result and divide by that. Is there some quicker method that I'm missing?

 

[Nathanael]

I'm guessing you need to find out how many nanometers are in a centimeter and then cube the result and divide by that. Is there some quicker method that I'm missing?

I would say no. Cubing the linear conversion factor is the most straightforward way to get the volumetric conversion factor.
If you find a quicker method, I will be tempted to call you a witch.

In this case it's (10^-7)³=10^-21

 

[Bystander]

Not recommended: but used --- look at prefixes; "milli-milli- is unit sp. gr.," anything larger (deci-, centi-) is less, and anything smaller (micro-, nano-) is more. If you've got other quick mental conversions for "mixed" prefixes, you can be halfway safe.

 

[DrPapper]

IGNORE THIS POST, I DID SOME REALLY MIXED UP STUFF, I'LL EDIT IT LATER BUT RIGHT NOW I HAVE TO LEAVE TO DO SOMETHING... LOL SORRY GUYS

Here is what I would do, which is ultimately the same the above response but more of a "step-by-step" answer. In time you'll get faster and more intuitive with this stuff and can combine or omit steps you know will cancel later.

_______________STEPS_____________________________________________________________________________________________

0. Use a conversion factor.

Unfortunately not every thing uses the same base measure, say inches to meters or Liters to cubic meters. So you'll need to memorize or look up conversions. Here we can see that D. and E. use a form of liters while the others use some form of meters. So use the conversion that 1L=1x10^-3m. Which means multiplying by the ratio of these two such that the undesired unit is canceled out. Since the two are equal to each other the numbers 1L/(1x10^-3) is the same as multiplying by 1. So is (1x10^-3)/1L

On E. then we'd use the last version of 1 which is (1x10^-3)/1L to multiply to 2.5L which would then give us the base unit we want. If you were doing D save the convert the number of the unit using step 1, and convert the prefix using step 2 then do this step 0.​

1. Convert the number given into scientific notation (disregard the 10^whatever for the unit prefix).

So for a. you'd get 1.3x10² dm (THE d on meters is the unit prefix I'm referring to)
​

2. Convert prefix into a 10^whatever for the unit system of choice (SI for example...) and multiply to the numerical part.

Again we'll use a. So dm=10^-1 which gets multiplied through to the 1.3x10². So 1.3x10² x 10^-1 = 1.3x10^2-1 = 1.3x10¹. Now my measurement of volume is in a standard of cubic meters and I'd do that to each answer. But if you start with it not in a "friendly base" you'll need some sort of conversion factor before doing the above steps.
​

 

[Nathanael]

Again we'll use a. So dm=10^-1 which gets multiplied through to the 1.3x10². So 1.3x10² x 10^-1 = 1.3x10^2-1 = 1.3x10¹. Now my measurement of volume is in a standard of cubic meters and I'd do that to each answer.

That's not right.
" dm=10^-1 " is what I called the "linear conversion factor" in my first post. To get what I called the "volumetric conversion factor" you have to cube it.
1 dm = 10^-1 m
1 cubic dm = 10^-3 cubic m

What you did was convert 130 decimeters to meters. That is not the same as converting 130 cubic decimeters to cubic meters.
The correct answer would be 1.3*10^2-3=1.3*10^-1

 

[DrPapper]

That's not right.
" dm=10^-1 " is what I called the "linear conversion factor" in my first post. To get what I called the "volumetric conversion factor" you have to cube it.
1 dm = 10^-1 m
1 cubic dm = 10^-3 cubic m

What you did was convert 130 decimeters to meters. That is not the same as converting 130 cubic decimeters to cubic meters.
The correct answer would be 1.3*10^2-3=1.3*10^-1

OMG what the heck did I write. LOL You're right. I need to go back and majorly fix that. For some reason I'm doing only linear conversions but kinda doing volume but adding it up to be a big confusing LIE. LOL Sorry guys...

 

[SteamKing]

You should also learn that 1 cubic decimeter = 1 liter (or litre, for the purists).

http://en.wikipedia.org/wiki/Litre

 

[Drakkith]

I would say no. Cubing the linear conversion factor is the most straightforward way to get the volumetric conversion factor.
If you find a quicker method, I will be tempted to call you a witch.

In this case it's (10^-7)³=10^-21

Wow, putting it as (10^-7)³ sure makes it easier to comprehend. Thanks for that. No more wordy things in long, broken structures to deal with. Just wonderful numbers!

Drakkith's new term of the day: Linear Conversion Factor.