Adiabatic law (no gain /heat loss) for the expansion of air is
PV^1.4=C,P(pressure) in lb/in^2 ,V(volume)in cubic inches, and C(constant). At a specific instant, the
pressure is 40 lb/in^2 and is increasing at the rate of 8 lb/in^2each second. If
C=5/16
, what is the rate of change of the volume at this instant?
V^1.4 dP/dt + 1.4P V^0.4 dV/dt = 0
now plug in your numbers to find dV/dt
To find the rate of change of the volume at a specific instant, we need to differentiate the given equation with respect to time.
The adiabatic law states that PV^1.4 = C, where P is the pressure, V is the volume, and C is a constant.
Differentiating both sides of the equation with respect to time (t), we get:
(dP/dt)V^1.4 + P * d(V^1.4)/dt = 0
Since the pressure is increasing at the rate of 8 lb/in^2 each second (dP/dt = 8), we can substitute this value into the equation:
(8)V^1.4 + P * d(V^1.4)/dt = 0
Now, we need to find the value of P at the specific instant. In the question, it is mentioned that the pressure is 40 lb/in^2 at this instant. Substituting this value into the equation:
(8)V^1.4 + (40) * d(V^1.4)/dt = 0
Next, we need to calculate the derivative term. The derivative of V^1.4 with respect to time can be found using the power rule of differentiation. The power rule states that if f(x) = x^n, then the derivative of f(x) with respect to x is df/dx = n * x^(n-1). Applying this rule:
d/dt(V^1.4) = 1.4 * V^(1.4 - 1) * dV/dt
= 1.4 * V^0.4 * dV/dt
Now, substitute this value back into the equation:
(8)V^1.4 + (40) * 1.4 * V^0.4 * dV/dt = 0
Simplifying further, we can divide the equation by V^1.4 to isolate the rate of change of the volume (dV/dt):
8 + 56 * V^(-1) * dV/dt = 0
To find the rate of change of the volume at this instant, we can rearrange the equation and solve for dV/dt:
56 * V^(-1) * dV/dt = -8
dV/dt = -8 * V / (56 * V^(-1))
Simplifying the expression:
dV/dt = -V^2 / 7
Now, we have the rate of change of the volume (dV/dt) in terms of V. The specific value of V at this instant is not provided, so we cannot find the exact rate of change without knowing the volume. However, using this equation, you can calculate the rate of change of the volume for any given value of V.