BOYLE\'S LAW FOR IDEAL GASES. If the temperature and the mass of a confined idea
ID: 2876282 • Letter: B
Question
BOYLE'S LAW FOR IDEAL GASES. If the temperature and the mass of a confined ideal agas are fixed, then PV=K, where P is the pressure and V is the volumn of the gas, and k is a constant. How fast is the volume of the gas changing at the moment that the pressure is 45 lb/in^2, the volume is 60 in^3, and the pressure is increasing at a rate of 3 lb/in^2 per minute? Give an exact number.
CYLINDER. A right circular cylinder's volume is shrinking at the rate of 15 cm^3/hr in such a way that its base radius is always twice its height. The cylinder retains a right circular cylindrical shape. How fast is the base radius changing when the height is 1 meter? Give an exact answer.
Explanation / Answer
1)PV=K
pressure is 45 lb/in^2, the volume is 60 in^3
45*60=K
=>K=2700
PV=2700
V=2700/P
differentiate with respect to time on both sides
dV/dt=(-2700/P2)dP/dt
pressure is increasing at a rate of 3 lb/in^2 per minute =>dP/dt =3
dV/dt=(-2700/452)*3
dV/dt=-4
volume of the gas is decreasing at 4 in^3 per minute
2)volume of cylinder ,V=r2h
base radius is always twice its height =>r =2h
=>h=r/2
volume of cylinder ,V=r2(r/2)
volume of cylinder ,V=(/2)r3
when the height is 1 meter , r=2*1 =2m =200 cm
differentiate with respect to time t .
dV/dt =(/2)3r2 dr/dt
volume is shrinking at the rate of 15 cm^3/hr => dV/dt =-15 ,r =200
-15 =(/2)3*2002 dr/dt
dr/dt =15/[(/2)3*2002]
dr/dt =-1/[4000]
base radius is decreasing at 1/[4000] cm/hour
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