# Homework Help: Very Confusing Units

1. Jun 11, 2006

### Richay

Metric unit conversions -

100 square kilometers to square centimeters

English units to metric units -

200 cubic yards to cubic centimeters

How would I solve that?

And does anybody know how I would enter my answers in the correct Unit Multiplier Form?

2. Jun 11, 2006

### vsage

How many centimeters are equal to one yard? I don't know the number off the top of my head but google says 91.44 cm / yard. How many cubic centimeters are in one cubic yard then? It would simply be the number of centimeters in one yard cubed. Also, I think you have one too many zeros for the 100 square km -> cm^2 conversion.

3. Jun 11, 2006

I don't know the process you are using to solve this. Maybe this way will be new to you, maybe it won't.

Write "100 square kilometers to square centimeters" out in algebra.

$$100\,km^2 = \lambda\,cm^2$$
$$100 \times 10^3 \,m^2 = \lambda \times 10^{-2} \,m^2$$

Notice $m^2$ cancels.

We now solve for $\lambda$:
$$\lambda = \frac{100\times 10^3}{10^{-2}}=100\times 10^{3+2}=100\times 10^5$$

Which is what you got.

Since you are looking for a multiplier (which I called $\lambda$ in this case) you should see that all the units will cancel. Therefore you should convert everything to the same units (notice how I dropped the $k$ by substituting in $10^3$) so that you can easily cancel and solve for the multiplier.

Last edited: Jun 12, 2006
4. Jun 12, 2006

### VietDao29

This is not correct. Notice that it's km2, not just km.
So let's go from 100 km2 to cm2.
100 km2 = 102 km2 = 104 hm2 = 106 dam2 = 108 m2 = 1010 dm2 = 1012 cm2.
-----------
1 yd = 91.44 cm, right?
So that means
1 yd3 = 91.443 cm3.
Just think about a cube of edge length 1yd (91.44 cm), it's volume is 1 yd3 or 91.44 x 91.44 x 91.44 = 91.443 cm3, right? (notice that it's edge length is 91.44 cm)
So 1 yd3 = 91.443 cm3
200 yd3 = ? cm3
Can you go from here? :)

5. Jun 12, 2006

### Ouabache

I also differ with FrogPad's solution for part (a)
Obviously there are various methods of arriving at the correct answer.
Here is my approach, as you go through it, it becomes intuitively obvious how to proceed.

Draw a picture of what you are starting with.
(a) draw a square with 1001/2 km. on each side.
So you only have 10 km per side of the square. (what is the area of this square?).

How to convert from km to cm. Whatever value in km lets call N.
(N km)(1000m/km)(100cm/m) notice what cancels; km and m
leaving N (103)(102) cm = N (105) cm
Recall N=10 in this example. So how many cm do you have on each side?
10(105)= 106 cm
Now you know the length of one side of the square in cm, what is the area of that square? (hint: value will be in cm2.

(b) English Units to Metric Units
Do it the same way.. Start by drawing a cube with 2001/3 yards on each side.
How do you convert from yards to cm? Try to use relations you already know. You will need one to go from English to Metric. I remember there are 2.54 cm. per inch.
Whatever value in yards lets call M. (M yds)(3ft/yd)(12in./ft)(2.54cm/in)
what cancels? (yds, ft and inches) leaving M (91.44)cm (same as what vsage found).

Recall in this example M=2001/3. So along one side of the cube you have
2001/3(91.44) cm. Now you know the length of one side of the cube in cms., what is the volume of that cube? (hint: answer is in cm3)

Last edited: Jun 12, 2006
6. Jun 12, 2006

### Integral

Staff Emeritus
Start from the basics.
convert km to meter we know the conversion factors:

1000$\frac m {km}$

and

100 $\frac {cm} m$

combine these to get

$$10^3 \frac m {km} X 10^2 \frac {cm} m = 10^5 \frac {cm} {km}$$

This is the conversion factor for converting km to cm, note the units indicate this.

Now to convert $km^2$ to $cm^2$ square the above conversion factor

$$( 10^5 \frac {cm} {km})^2 = 10^{10} \frac {cm^2} {km^2}$$

now since you are converting 100 km2 to cm2 simply multiply by the conversion factor.

100 $km^2$ * $10^{10} \frac {cm^2} {km^2}= 10^{12} cm^2$

note that the units are correct.

Last edited: Jun 12, 2006
7. Jun 12, 2006

### dav2008

I don't know why you guys are setting up algebraic expressions or drawing squares.

Simple factor-label method is sufficient for converting between units.

Example:
(Convert 10. cubic feet to cubic centimeters)
$$\frac{10. \ ft^3}{1} \ \cdot \ \large{(}\frac{12 \ in}{1 \ ft})^3 \ \cdot \ (\frac{2.54 \ cm}{1 \ in})^3 \ = \ (10 \ \times \ 12^3 \ \times \ 2.54^3) \ cm^3 = \ 2.8 \cdot 10^5 \ cm^3$$

Notice how when you distribute the cubes, the units will cancel out leaving you with cm3

Last edited: Jun 12, 2006
8. Jun 15, 2006

-----------

My internet went down before I had a chance to preview this post. I actually forgot about it. However, where is my error?

Last edited: Jun 15, 2006
9. Jun 15, 2006

### Ouabache

As I mentioned earlier, there are various approaches that will arrive at the correct answer. The system I demonstrated employs a common technique of http://oakroadsystems.com/math/convert.htm [Broken]). I use the geometrical constructs (squares, cubes) for easier visualization.

Note for the 1st part VietDao29, myself and Integral all come to the same solution. For the 2nd part VietDao29 and mine agree and applying the dimensional analysis method to dav2008's example, he and I come to the same result. (see below)

For dav2008's example: Converting 10ft3 to cm3,
we may draw a cube with 101/3ft on each side.
Let M = 101/3. To convert ft to cm: (1ft) (12in/ft)(2.54cm/in) = 30.48cm
Each side (s) of the cube would be M(30.48) = 101/3 (30.48) cm.
Volumecube = s3 = [101/3(30.48)]3 = 2.83 x 105 cm3

Since we have more that one valid method, I would choose the one that makes the most sense to you or better yet, use more than one method to double check yourself.

Last edited by a moderator: May 2, 2017
10. Jun 15, 2006

### Ouabache

$$100\,km^2 = \lambda\,cm^2$$

is not equivalent to: $$100 \times 10^3 \,m^2 = \lambda \times 10^{-2} \,m^2$$

It is:$$100 \times [10^3 \,m]^2 = \lambda \times [10^{-2} \,m]^2$$

Last edited: Jun 15, 2006
11. Jun 15, 2006