Question

4.) calculate the value of r_t, i_t, i_r1, i_r2, i_r3. use units and proper engineering notation for full credit.
12v vt
8.2kω r1
6.8kω r2
4.7kω r3

Explanation:

Step1: Calculate equivalent resistance of parallel - connected R1 and R2

The formula for two resistors in parallel $R_{eq1}=\frac{R_1\times R_2}{R_1 + R_2}$. Here, $R_1 = 8.2\times10^{3}\Omega$ and $R_2=6.8\times10^{3}\Omega$.
$R_{eq1}=\frac{8.2\times10^{3}\times6.8\times10^{3}}{8.2\times10^{3}+6.8\times10^{3}}=\frac{55.76\times10^{6}}{15\times10^{3}}\Omega = 3.717\times10^{3}\Omega$

Step2: Calculate total resistance $R_T$

$R_T$ is the sum of the equivalent resistance of the parallel - part and $R_3$. Since $R_3 = 4.7\times10^{3}\Omega$ and $R_{eq1}=3.717\times10^{3}\Omega$, then $R_T=R_{eq1}+R_3=(3.717 + 4.7)\times10^{3}\Omega=8.417\times10^{3}\Omega\approx8.42\times10^{3}\Omega$

Step3: Calculate total current $I_T$

Using Ohm's law $I_T=\frac{V_T}{R_T}$, where $V_T = 12V$ and $R_T=8.417\times10^{3}\Omega$. So $I_T=\frac{12}{8.417\times10^{3}}A\approx1.43\times10^{- 3}A$

Step4: Calculate current through $R_1$

The voltage across the parallel - part $V_{parallel}=I_T\times R_{eq1}$. $V_{parallel}=1.43\times10^{-3}A\times3.717\times10^{3}\Omega\approx5.32V$. Then, using Ohm's law for $R_1$, $I_{R1}=\frac{V_{parallel}}{R_1}=\frac{5.32V}{8.2\times10^{3}\Omega}\approx6.49\times10^{-4}A$

Step5: Calculate current through $R_2$

Using Ohm's law for $R_2$, $I_{R2}=\frac{V_{parallel}}{R_2}=\frac{5.32V}{6.8\times10^{3}\Omega}\approx7.82\times10^{-4}A$

Step6: Calculate current through $R_3$

$I_{R3}=I_T = 1.43\times10^{-3}A$

Answer:

$R_T\approx8.42\times10^{3}\Omega$, $I_T\approx1.43\times10^{-3}A$, $I_{R1}\approx6.49\times10^{-4}A$, $I_{R2}\approx7.82\times10^{-4}A$, $I_{R3}\approx1.43\times10^{-3}A$