In a circuit with resistance 10 ohms, power 𝑃 (measured in watts) is equal to 10𝐼2 , where 𝐼 is the current measured in amperes. When 𝐼=6 amperes, what is the rate of change of 𝑃 with respect to 𝐼 ?
To find the rate of change of power with respect to current, we need to calculate the derivative of power with respect to current.
Given that power, P, is equal to 10I^2, we can find the derivative using the power rule of differentiation. The power rule states that if we have a function of the form f(x) = ax^n, where a and n are constants, then its derivative is given by f'(x) = nax^(n-1).
Thus, we can differentiate the expression for power with respect to current, I, using the power rule. Taking the derivative of P = 10I^2, we get:
dP/dI = 2 * 10I^(2-1)
= 20I
Now, we can substitute the given value of current, I = 6 amperes, into the derivative to find the rate of change of power with respect to current:
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.
P = I^2 R = 10I^2
dP/dI = 2I
so plug in your numbers