Question

Problem 3. (15 points) An electric current flowing in a circuit (consisting of a battery of $V$ volts and a resistor of $R$ ohms connected in series), satisfies $\frac{dI}{dt} = -k(I - b)$ for some constants $k$ and $b$ with $k > 0$. Initially, $I(0) = 0$ and $I(t)$ approaches a maximum level $\frac{V}{R}$ as $t \to \infty$. (a) What is the value of $b$? b = $\frac{V}{R}$ (b) Find a formula for $I(t)$ in terms of $k$, $V$, and $R$. $I(t) = e^{\frac{kV}{Rt}}$ (c) What percent of its maximum value does $I(t)$ reach at time $t = \frac{3}{k}$? 98.74 % Note: You can earn partial credit on this problem.

          Problem 3.
(15 points)
An electric current flowing in a circuit (consisting of a battery of $V$ volts and a resistor of $R$ ohms connected in series), satisfies
$\frac{dI}{dt} = -k(I - b)$
for some constants $k$ and $b$ with $k > 0$. Initially, $I(0) = 0$ and $I(t)$ approaches a maximum level $\frac{V}{R}$ as $t \to \infty$.
(a) What is the value of $b$?
b = $\frac{V}{R}$
(b) Find a formula for $I(t)$ in terms of $k$, $V$, and $R$.
$I(t) = e^{\frac{kV}{Rt}}$
(c) What percent of its maximum value does $I(t)$ reach at time $t = \frac{3}{k}$?
98.74
%
Note: You can earn partial credit on this problem.

Problem 3.
(15 points)
An electric current flowing in a circuit (consisting of a battery of V volts and a resistor of R ohms connected in series), satisfies
(dI)/(dt) = -k(I - b)
for some constants k and b with k > 0. Initially, I(0) = 0 and I(t) approaches a maximum level (V)/(R) as t →∞.
(a) What is the value of b?
b = (V)/(R)
(b) Find a formula for I(t) in terms of k, V, and R.
I(t) = e^(kV)/(Rt)
(c) What percent of its maximum value does I(t) reach at time t = (3)/(k)?
98.74
%
Note: You can earn partial credit on this problem.

Added by Sean B.

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