Calculus III problem

[eight10]

The relationship between the pressure P, volume V and temperature T of a certain gas is given by PV= 12T. If the pressure is decreasing at the constant rate of 3 psi/min and the temperature is constantly increasing at 4°K per minute, find the rate at which the volume is changing when P = 10 psi and T = 298°K.


I can't seem to make any progress on this one, not sure how to start it. Obviously a rate of change problem but with 2 variables.

Thank you ahead of time.

 

[tkhunny]

Let's play like everthing is a function of the Temperature.

P(T)*V(T) = 12*T

The implicit derivative, using the differential notaion, gives:

P(T)*dV + V(T)*dP = 12*dT

"the pressure is decreasing at the constant rate of 3 psi/min"

dP = -3 psi / min

"the temperature is constantly increasing at 4°K per minute"

dT = +4°K / min

"find the rate at which the volume is changing"

Find dV

when P = 10 psi and T = 298°K.

There is a tiny problem. What is it and how do you fix it?

 

[BigGlenntheHeavy]

PV = 12T implies V = 12T/P

\(\displaystyle Hence, \ \frac{dV}{dt} \ = \ \frac{(12P)\frac{dT}{dt}-(12T)\frac{dP}{dt}}{P^2}\)

Can you take it from here?

Note: This reverts to Boyle's Law, hence do you have to convert some or all units, as my chemistry is lacking.

 

[eight10]

Awesome thank you. I also solved it another way, but thank you for confirming my answer.

 

[BigGlenntheHeavy]

As always, eight10, there is more than one way to skin a cat.

I and tkhunny showed you two, and you had a third.