I heart variations

[Fatal Dawn]

I absolutely adore variations. I'm not being sarcastic because it's the area I have picked up easily and what I'm best in. I love doing them. Here's an example - Boyle's Law states that the pressure of a sample of gas is directly proportional to the temperature and inversely proportional to the volume.

In other words.



P = kT / v

take a 100L gas exerting a pressure of 33.2 kPa at an absolute temperature measurement of 400 Kelvin

k = Pv/ T = (100)(33.2) / (400) = 8.3 constant of proportionality

the pressure of the gas if temp increased to 500K and the volume decreased to 80L

P = (8.3)(500)/(80) = 51.875kPa



Makes sense. Higher temps and lower volume would equal a greater pressure.





I would appreciate anyone who could help me solve Kepler's Third law of planetary motion. This one has eluded me for so long.



The square of a planet's period is directly proportional to the cube of its average distance from the sun. So..

T[sup]2 [/sup]= kd[sup]3[/sup]

k= T[sup]2[/sup]/d[sup]3[/sup]



Earth's planetary period is of course about 365 days and the average distance is 93 million miles. Find the constant.

k = (365)[sup]2[/sup]/(93,000,000)[sup]3[/sup] = 1.65 * 10[sup]-19[/sup]



To solve for Neptune which is around 2.8 billion miles from the sun.

T[sup]2[/sup] = 1.65 * 10[sup]-19 [/sup](2.79 * 10[sup]9[/sup]) = 4.6035 * 10[sup]10[/sup]



I think I screwed up here

 

[Fatal Dawn]

I can't solve the variation for Neptune.

 

[Jazzy]

I can't solve the variation for Neptune.

Very interesting thread and I'm sure you'll solve it! Let us know when you do!

 

[The Dragon Master]

I applaude you for being able to do this. My head hurts looking at it >_<

 

[Nebulous's iconNebulous]

Look a butterfly.

 

[Fatal Dawn]

I found what I did wrong and I think I have calculated the answer now.



k [font=tahoma, helvetica, arial, sans-serif]= (365)[sup][font=tahoma, helvetica, arial, sans-serif]2[/sup][font=tahoma, helvetica, arial, sans-serif]/(93,000,000)[sup]3[/sup]= 1.07 * 10[sup]23[/sup]

[font=tahoma, helvetica, arial, sans-serif](133225)/(8.04357) * 10[sup][font=tahoma, helvetica, arial, sans-serif]23 [/sup][font=tahoma, helvetica, arial, sans-serif]= 1657 * 10[sup][font=tahoma, helvetica, arial, sans-serif]23[/sup]



[font=tahoma, helvetica, arial, sans-serif](1657 * 10[sup][font=tahoma, helvetica, arial, sans-serif]23[/sup][font=tahoma, helvetica, arial, sans-serif])[font=tahoma, helvetica, arial, sans-serif](2.8 * 10[sup]9[/sup][font=tahoma, helvetica, arial, sans-serif])[sup]3[/sup] or

[font=tahoma, helvetica, arial, sans-serif]3.64 * 10 [sup]54[/sup]

√[font=tahoma, helvetica, arial, sans-serif]3.64 * 10 [sup]54 [/sup][font=arial, sans-serif]≈[sup] [/sup][font=tahoma, helvetica, arial, sans-serif] 1.907878403 * 10[sup]27[/sup]





So Neptune's maximum revolutionary period is roughly equal to 190 Earth years.

 

[Evil Eye]

k [font=tahoma, helvetica, arial, sans-serif]= (365)[sup][font=tahoma, helvetica, arial, sans-serif]2[/sup][font=tahoma, helvetica, arial, sans-serif]/(93,000,000)[sup]3[/sup]= 1.07 * 10[sup]23[/sup]

There's no way that's correct

Numbers get smaller as the number you divide by gets bigger and 93e6^3 is a lot bigger than 365^2.

Anyway, if you've seen my response you should know the answer by now (unless I screwed up

).



For anyone interested T = sqrt(3597075000) (~ 59975.62).

 

[Fatal Dawn]

Originally I didn't sqrt T. So let's start again from the constant. We have:



[font=tahoma, helvetica, arial, sans-serif]k=(365)[sup]2[/sup][font=tahoma, helvetica, arial, sans-serif]/(93,000,000)[sup]3[/sup][font=tahoma, helvetica, arial, sans-serif] which is [font=tahoma, helvetica, arial, sans-serif]1.65629192*10^-19

[font=tahoma, helvetica, arial, sans-serif]T[sup]2[/sup] = k(1.66e-19)d(2.79e9)^3 = 3605128074 ?



[font=tahoma, helvetica, arial, sans-serif]sqrt: 3605128074 = 60042.71874

[font=tahoma, helvetica, arial, sans-serif]60042.71874 / 365 would equal the 164.5 Earth years.



[font=tahoma, helvetica, arial, sans-serif]Pretty close to NASA's estimation of 164.8 Earth years

 

[Evil Eye]

Correct, but if you don't round off in between you get a nicer number

 

[Fatal Dawn]

Awesome, thanks for your help EE. This one equation has been bothering me to no end.

 

[Evil Eye]

No problem.

What else do you guys have to do?

 

[Fatal Dawn]

I'm working on functions now. Solving, finding domain and range, and graphing.

 

[Evil Eye]

graphing.

You have to draw the things? I hate that.

Don't you have a calculator for that?

 

[Fatal Dawn]

I hate it too. We're not allowed to use graphing calculators.

 

[Evil Eye]

Poor you!