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 670 cubic centimeters and the pressure is 97 kPa and is decreasing at a rate of 7 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?
PV^1.4 = C
V^1.4 P' + 1.4PV^.4 V' = 0
(60^1.4)(-7) + (1.4)(670)V' = 0
V' = 2.3 cc^3/min
Where did the 60 come from in your equation?
To find the rate at which the volume is increasing at the given instant, we can use the chain rule from calculus. The chain rule tells us that if a variable (in this case, time) affects another variable (in this case, volume), then the rate of change of the dependent variable (volume) with respect to the independent variable (time) can be found by multiplying the rate at which the independent variable is changing (pressure) with the rate at which the dependent variable changes with respect to the independent variable (rate at which volume changes with respect to pressure).
Let's define the variables:
- V: Volume (in cubic centimeters)
- P: Pressure (in kilopascals)
- t: Time (in minutes)
We are given:
- Initial volume: V = 670 cm^3
- Initial pressure: P = 97 kPa
- Rate of change of pressure: dP/dt = -7 kPa/min (pressure is decreasing)
We need to find:
- Rate of change of volume: dV/dt (volume is increasing)
The given relationship between pressure and volume is: PV^1.4 = C
To find the rate of change of volume with respect to time, we differentiate both sides of the equation implicitly with respect to time (using the product rule and the chain rule):
d(PV^1.4)/dt = dC/dt = 0 (since C is a constant)
Differentiating the left side:
d(PV^1.4)/dt = P * dV^1.4/dt + V^1.4 * dP/dt
Now, let's solve for dV/dt:
P * dV^1.4/dt = -V^1.4 * dP/dt
Divide both sides by P:
dV^1.4/dt = (-V^1.4 * dP/dt) / P
Take the derivative of V^1.4 with respect to V using the power rule:
1.4 * V^0.4 * dV/dt = (-V^1.4 * dP/dt) / P
Simplify:
dV/dt = (-V^1.4 * dP/dt) / (1.4 * V^0.4 * P)
Now, substitute the given values into the equation:
dV/dt = (-(670^1.4) * (-7)) / (1.4 * (670^0.4) * 97)
Simplifying further, we get:
dV/dt ≈ 1.51 cm^3/min (rounded to two decimal places)
Therefore, at this instant, the volume is increasing at a rate of approximately 1.51 cubic centimeters per minute.