When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4=C where C is a constant. Suppose that at a certain instant the volume is 470 cubic centimeters and the pressure is 89 kPa and is decreasing at a rate of 11 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?
To find the rate at which the volume is increasing, we can use the equation PV^1.4 = C and differentiate it implicitly with respect to time.
Differentiating both sides of the equation with respect to time gives us:
d(PV^1.4)/dt = dC/dt
Using the product rule on the left-hand side, we get:
V^1.4 dP/dt + 1.4P(V^0.4)(dV/dt) = dC/dt
We are given the pressure and its rate of change, so we can substitute those values into the equation:
470^1.4(-11) + 1.4(89)(470^0.4)(dV/dt) = 0
Simplifying this equation gives:
-11(470^1.4) + 1.4(89)(470^0.4)(dV/dt) = 0
Now we can solve for dV/dt, the rate at which the volume is increasing:
1.4(89)(470^0.4)(dV/dt) = 11(470^1.4)
Multiply both sides by (1/1.4)(1/89)(1/470^0.4) to isolate dV/dt:
dV/dt = (11(470^1.4))/ (1.4(89)(470^0.4))
Simplifying further,
dV/dt = (11/1.4) * 470^(1.4 - 0.4)
Calculating the above expression gives the rate at which the volume is increasing at that instant.
To find the rate at which the volume is increasing, we need to use the given information and differentiate the equation PV^1.4 = C with respect to time.
Given:
Initial volume, V = 470 cubic centimeters
Initial pressure, P = 89 kPa
Rate of change of pressure, dP/dt = -11 kPa/minute (negative since pressure is decreasing)
Differentiating the equation PV^1.4 = C implicitly with respect to time t using the product rule:
d(PV^1.4)/dt = dC/dt
Let's differentiate each term separately:
For the term PV^1.4:
Using the product rule:
d(PV^1.4)/dt = (dP/dt)V^1.4 + P * d(V^1.4)/dt
To find d(V^1.4)/dt, differentiate V^1.4 with respect to V:
d(V^1.4)/dV = 1.4V^(1.4-1) = 1.4V^0.4 = 1.4√V
Now, differentiate V^1.4 with respect to time:
d(V^1.4)/dt = (d(V^1.4)/dV) * (dV/dt) = 1.4√V * dV/dt
Substituting this back into the expression:
d(PV^1.4)/dt = (dP/dt)V^1.4 + P * (1.4√V * dV/dt)
Since PV^1.4 = C, we can substitute C into the equation:
(dP/dt)V^1.4 + P * (1.4√V * dV/dt) = dC/dt
Now let's substitute the given values:
(-11 kPa/minute) * (470 cm^3)^1.4 + (89 kPa) * (1.4√470 cm^3) * (dV/dt) = 0
Simplifying the equation:
-11 * 470^1.4 + 1.4 * 89 * √470 * (dV/dt) = 0
You can now solve for dV/dt, the rate at which the volume is increasing, by rearranging the equation:
1.4 * 89 * √470 * (dV/dt) = 11 * 470^1.4
(dV/dt) = (11 * 470^1.4) / (1.4 * 89 * √470)
Now, you can calculate this expression using a calculator to find the rate at which the volume is increasing in cubic centimeters per minute.