How do go about this problem after finding dp/dt in terms of and dV/dt?

Question

At a certain moment a sample of gas obeying Boyle's law occupies a volume of 1000 in3 at a pressure of 10 lb/in2. If this gas is being compressed isothermally at the rate of 12 in3/min, find the rate at which the pressure is increasing at the instant when the volume is 600 in3

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Answer 1 · 2018-03-23T02:25:59

#(dp)/dt=1/3# #lb"/""in"^2"/"min#

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Boyle's Law states:

#PV=C#

Let's differentiate the equation:

#P(dv)/dt+V(dp)/dt=0#

#V(dp)/dt=-P(dv)/dt#

Let's divide both sides by #V#:

#(dp)/dt=-P/V(dv)/dt#

We know another form of Boyle's Law as follows:

#P_iV_i=P_fV_f# where the subscripts #i# and #f# indicate initial and final:

#P_i=10#

#V_i=1000#

#V_f=600#

#P_f=(P_iV_i)/V_f=((10)(1000))/600=50/3#

#(dp)/dt=-(50/3)/600(-12)=50/1800*12=600/1800=1/3# #lb"/""in"^2"/"min#