Related Rates

[asimon2005]

Calculus: Early Transcendentals 6th edition

Related Rates

#32. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.4=C, where C is a constant. Suppose that at a certain instant the volume is 400cm^3 and the pressure is 80kPA and is decreasing at a rate of 10kPA/min. At what rate is the volume increasing at this instant?

 

[BigGlenntheHeavy]

Assuming no conversion of units is necessary, we have;

$\displaystyle Given: \ PV^{1.4} \ = \ C, \ C \ a \ constant, \ V \ = \ 400, \ P \ = \ 80, \ and \ \frac{dP}{dt} \ = \ -10, \ find \ \frac{dV}{dt}.$

$\displaystyle \frac{dP}{dt}V^{1.4}+P(1.4)V^{.4}\frac{dV}{dt} \ = \ 0, \ P(1.4)V^{.4}\frac{dV}{dt} \ = \ -\frac{dP}{dt}V^{1.4}$

$\displaystyle \frac{dV}{dt} \ = \ \frac{-\frac{dP}{dt}V^{1.4}}{P(1.4)V^{.4}} \ = \ \frac{10V}{(1.4)P} \ = \ \frac{(10)(400)}{(1.4)(80)}$

$\displaystyle Hence, \ \frac{dV}{dt} \ = \ \frac{4000}{112} \ = \ \frac{250}{7} \ = \ 35.7 \ \frac{cm^{3}}{min.}$

 

[asimon2005]

Thanks for the help.