Q. 114 - Precalculus Enhanced with Graphing Utilities

Question

The equation governing the amount of current I (in amperes) after time t (in microseconds) in a single RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E (in volts) is

$I=\frac{E}{R}{e}^{-t/\left(RC\right)}$

Part (a): If $E = 120 volts, R = 2000 ohms, and C = 1.0 microfarad$, how much current $I_1$ is flowing initially ($t = 0$)? After $1000$microseconds? After $3000$ microseconds?

Part (b): What is the maximum current?

Part (c): Graph the function $I=I_1\left(t\right)$, measuring I along the y-axis and t along the x-axis.

Part (d): If $E = 120 volts, R = 1000 ohms, and C = 2.0 microfarads$, how much current $I_2$ is flowing initially? After $1000$ microseconds? After $3000$ microseconds?

Part (e): What is the maximum current?

Part (f): Graph the function $I=I_2\left(t\right)$ on the same coordinate axes as $I_1\left(t\right)$.

Step-by-Step Solution

Answer

Part (a): The current flowing after $0microseconds$ will be $0.06amp$.

The current flowing after $0.5microseconds$ will be $0.036amp$.

The current flowing after $3000microseconds$ will be $0.013amp$.

Part (b): The maximum current is $0.06amp$,

Part (c): The graph of the function $I=I_1\left(t\right)$ is given below,

Part (d): The current flowing after $0 microseconds$ will be $0.12amp$.

The current flowing after  $1000microseconds$ will be $0.072amp$.

The current flowing after $3000microseconds$ will be $0.027amp$.

Part (e): The maximum current is $0.12amp$.

Part (f): The graph of the function $I=I_2\left(t\right)$ is given below,

Part (a) Step 1. Determine the amount of current after t = 0 .

Consider the given function,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=2000, \\ C=1, \\ t=0 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{2000}{e}^{-0/\left(2000\times 1\right)} \\ =0.06amp \end{gathered}$$

Part (a) Step 2. Determine the amount of current after t = 1000 .

Consider the given function,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=2000, \\ C=1, \\ t=1000 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{2000}{e}^{-1000/\left(2000\times 1\right)} \\ =0.036amp \end{gathered}$$

Part (a) Step 3. Determine the amount of current after t = 3000 .

Consider the given function,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=2000, \\ C=1, \\ t=1000 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{2000}{e}^{-3000/\left(2000\times 1\right)} \\ =0.013amp \end{gathered}$$

Part (b) Step 1. Determine the maximum current.

Assume t to be the time when the current is maximum.

Since it is decreasing function then current is maximum at $t=0$.

$$\begin{gathered} I=\frac{120}{2000}{e}^{-0/\left(2000\times 1\right)} \\ =0.06amp \end{gathered}$$

Part (c) Step 1. Plot the function.

Consider the given question,

$\left(0,0.06\right),\left(1000,0.036\right),\left(3000,0.013\right)$

Plot the points on the graph,

Part (d) Step 1. Determine the amount of current after t = 0 .

Consider the given question,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=1000, \\ C=2, \\ t=0 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{1000}{e}^{-0/\left(1000\times 2\right)} \\ =0.12 amp \end{gathered}$$

Part (d) Step 2. Determine the amount of current after t = 1000 .

Consider the given function,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=1000, \\ C=2, \\ t=1000 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{1000}{e}^{-1000/\left(1000\times 2\right)} \\ =0.072 amp \end{gathered}$$

Part (d) Step 3. Determine the amount of current after t = 3000 .

Consider the given question,

$$\begin{gathered} I=\frac{E}{R}{e}^{-t/\left(RC\right)}, \\ E=120, \\ R=1000, \\ C=2, \\ t=3000 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} I=\frac{120}{1000}{e}^{-3000/\left(1000\times 2\right)} \\ =0.027 amp \end{gathered}$$

Part (e) Step 1. Determine the maximum current.

Assume t to be the time when the current is maximum.

Since it is decreasing function then current is maximum at $t=0$.

$$\begin{gathered} I=\frac{120}{1000}{e}^{-0/\left(1000\times 2\right)} \\ =0.12 amp \end{gathered}$$

Part (f) Step 1. Plot the function.

Consider the given question,

$\left(0,0.12\right),\left(1000,0.072\right),\left(3000,0.056\right)$

Plot the points on the graph,