A. In a circuit with resistance 10 ohms, power 𝑃 (measured in watts) is equal to 10𝐼^2 , where 𝐼 is the current measured in amperes. When 𝐼=6 amperes, what is the rate of change of 𝑃 with respect to 𝐼 ?
B. If we additionally know that when 𝐼=6 , the current is increasing at 0.2 amperes per minute, at what rate (with respect to time) is power increasing at that moment?
C. What units is your answer measured in?
P = 10I^2
dP/dt = 2I dI/dt
again, plug in your numbers
units? just look at the units in the fraction that is the derivative
Unfortunately this is my weakest topic in Calculus so far, I'm still not able to find the correct answer.
I used product rule to break up 10*I^2
Should I even be doing this? Please elaborate further if possible. (Something a 5th grader could comprehend haha)
A. To find the rate of change of power with respect to current, we need to take the derivative of the power function with respect to current. In this case, the power function is given by 𝑃 = 10𝐼^2.
Taking the derivative, we have:
𝑑𝑃/𝑑𝐼 = 2 * 10𝐼^(2-1) = 20𝐼
Now we can substitute the value 𝐼=6 amperes into the derivative expression to find the rate of change of power:
𝑑𝑃/𝑑𝐼 = 20(6) = 120 watts per ampere
Therefore, the rate of change of power with respect to current when 𝐼=6 amperes is 120 watts per ampere.
B. To find the rate at which power is increasing with respect to time when 𝐼=6 amperes, we need to take into account the rate at which the current is changing with respect to time. We are given that the current is increasing at a rate of 0.2 amperes per minute.
Since power is a function of current, we can use the chain rule to express the rate of change of power with respect to time:
𝑑𝑃/𝑑𝑡 = 𝑑𝑃/𝑑𝐼 * 𝑑𝐼/𝑑𝑡
We already know the rate of change of power with respect to current from part A, which is 120 watts per ampere.
Now we need to find the rate of change of current with respect to time. Given that the current is increasing at a rate of 0.2 amperes per minute, we have:
𝑑𝐼/𝑑𝑡 = 0.2 amperes per minute
Substituting these values into the chain rule equation, we get:
𝑑𝑃/𝑑𝑡 = 120 * 0.2 = 24 watts per minute
Therefore, the rate at which power is increasing at that moment is 24 watts per minute.
C. The units of the answer in both parts A and B are watts per ampere (W/A) for the rate of change of power with respect to current, and watts per minute (W/min) for the rate at which power is increasing with respect to time.
read up on the chain rule
If P is a function of I, and I is a function of t, then
dP/dt = dP/dI * dI/dt