The power P (in W) generated by a particular windmill is given by P = .015 V^3 where V is the velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is 5.7 mph.
P = 0.015 V^3
dP/dV = 0.045 V^2 (in units of Watts per mph)
Plug V = 5.7 mph into the above formula to get the rate of change with respect to V.
Well, if we plug in V = 5.7 mph into the formula, we get dP/dV = 0.045 x (5.7)^2, which comes out to be...wait for it...0.884175 Watts per mph! That's a pretty precise rate of change, don't you think? It's almost like measuring ounces of laughter per minute.
Let's plug in the given velocity of 5.7 mph into the formula for the rate of change:
dP/dV = 0.045 V^2
Substituting V = 5.7:
dP/dV = 0.045 (5.7)^2
= 0.045 * 32.49
= 1.46205
Therefore, the instantaneous rate of change of power with respect to velocity when the velocity is 5.7 mph is approximately 1.46205 Watts per mph.
To find the instantaneous rate of change of power with respect to velocity, we need to take the derivative of the formula for power with respect to velocity.
Given that the formula for power is P = 0.015V^3, where V is the velocity of the wind in mph, we first differentiate the formula with respect to V.
We can use the power rule of differentiation, which states that if we have a function of the form f(x) = ax^n, then the derivative f'(x) is given by f'(x) = nax^(n-1).
In this case, a = 0.015 and n = 3, so we have P = 0.015V^3. Differentiating with respect to V, we have dP/dV = 0.045V^2.
So, the derivative of power with respect to velocity is dP/dV = 0.045V^2, expressed in units of Watts per mph.
To find the instantaneous rate of change of power with respect to velocity when the velocity is 5.7 mph, we simply substitute V = 5.7 mph into the derivative formula.
Plugging V = 5.7 mph into the formula dP/dV = 0.045V^2, we have:
dP/dV = 0.045(5.7)^2
Calculating this expression, we find:
dP/dV ≈ 0.91435
Therefore, the instantaneous rate of change of power with respect to velocity when the velocity is 5.7 mph is approximately 0.91435 Watts per mph.