hlilly2413 one year ago How many miles deep would a mole of popcorn kernels have to be in order to cover the continental US? Assume that the US is a rectangle with dimensions of 3000 miles by 1000 miles and that 90 kernels occupy a volume of 1 cubic inch. (1 mole of kernels = 6.022 * 10^23).

1. anonymous

No problem!

2. greatlife44

@Shalante nicely explained

3. hlilly2413

$\frac{ 6.022\times10^{23}kernals }{ 1 mole } \times \left( \frac{ 1 inches ^{3} }{ 90 kernals } \right)\times \left( \frac{ 1 miles ^{3} }{ 63360^{3} } \right)\times \left( \frac{ 1 }{ 3000miles } \right)$

4. hlilly2413

then $\left( \frac{ 1 }{ 1000 miles } \right)$

5. hlilly2413

63,360^3 inches ^3

6. Jhannybean

Where did the 63360 come from and what are the units?

7. anonymous

I need to go for now. Will be back later tonight and hopefully I can solve it. Not sure if my step that I deleted were correct though,

8. Jhannybean

Repost your steps @Shalante so i have something to work with, later!

9. hlilly2413

@Jhannybean the 63,3660 is how many inches are in a mile. The unit is cubic inches.

10. hlilly2413

First I tried to figure out how many kernels fit in 1 cubic mile. *The 63,360 came from the number of inches in one mile. If we are working with cubic miles then we need to convert from cubic inches. $\frac{ 90 kernels }{ inches ^{3} } \times \frac{ 63,360^{3}inches ^{3} }{ 1 mile ^{3} }= (90)(63.360)^{3}kernels per cubic mile.$ Now, since we have one mole of kernels, we need to figure out how many cubic miles one mole will fill. So find the volume I calculated: $\frac{ 6.022\times 10^{23} kernels }{ 90 \left( 63,360 \right)^{3}kernels per cubic mile }= 26305874.02 cubic miles$ Finally, since we know the length, the width, and now the total volume I performed the following equation to find the number of miles deep the mole of popcorn kernels would have to be in order to cover the continental US (providing it is shaped like a rectangle). $height = \frac{ volume }{ length \times weight }$ That's essentially what I did. However, since it is a dimensional analysis question I know everyone likes the answer to be in a row where everything cancels out. But there. I don't know if it is correct and if it isn't please find my error. :) (and then tell me because I need to fix it)

11. anonymous

Ok, this problem is actually not bad. I was not thinking hard enough. (**** me) Volume of a rectangular prism=whl Depth=height We know its w*l is 3000 miles*1000 miles=3,000,000 miles Volume of a rectangular prism with 1 mol of kernel=3000 miles*1000miles*h \$\frac{ 1mol of kernel }{ 1 }\times \frac{6.023\times10^{23} kernel}{ 1mol of kernel }\times \frac{ 1inches^3 }{ 90 kernels }\times \frac{ 1.58\times 10^{-5} miles }{ 1inch }\times \frac{1.58\times10^{-5} miles }{ 1inch }\times \frac{ 1.58\times 10^{-5} miles }{ 1inch }$=??miles^3 Solve for h