Convert English & Metric Units: 200 cubic yards to cubic cm

[Richay]

Metric unit conversions -

100 square kilometers to square centimeters

Wouldn't the answer be 10000000?

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?

 

[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.

 

[FrogPad]

Metric unit conversions -

100 square kilometers to square centimeters

Wouldn't the answer be 10000000?

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.

 

[VietDao29]

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

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.
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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? :)

 

[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 100^1/2 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 let's 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 200^1/3 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 let's 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=200^1/3. So along one side of the cube you have
200^1/3(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)

 

[Integral]

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.

 

[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

 

[FrogPad]

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.

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

 

[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 ). 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 10^1/3ft on each side.
Let M = 10^1/3. To convert ft to cm: (1ft) (12in/ft)(2.54cm/in) = 30.48cm
Each side (s) of the cube would be M(30.48) = 10^1/3 (30.48) cm.
Volume_cube = s³ = [10^1/3(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.

 

[Ouabache]

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

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

 

[FrogPad]

Woops.
Well the concept is there. Partial credit? :)