Question

A 6 ohm, 5 ohm, and 8 ohm resistor are connected in parallel with a 30 V battery. What is the total
resistance of the entire circuit and the current through the 8 ohm resistor?
Answers below:
Total Resistance = 19 ohms. Current = 3.75 A.

Total Resistance = 19 ohms. Current = 1.58 A.

Total Resistance = 2.03 ohms. Current = 14.78 A.

Total Resistance = 2.03 ohms. Current = 3.75 A.

Answer 1

To find the total resistance of resistors in parallel and the current through the 8-ohm resistor, we can use the following formulas:

1. The formula for total resistance \( R_t \) in a parallel circuit is given by:

\[
\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\]

where \( R_1 = 6 \) ohms, \( R_2 = 5 \) ohms, and \( R_3 = 8 \) ohms.

First, calculate \( R_t \):

\[
\frac{1}{R_t} = \frac{1}{6} + \frac{1}{5} + \frac{1}{8}
\]

Finding a common denominator (which is 120):

\[
\frac{1}{6} = \frac{20}{120}, \quad \frac{1}{5} = \frac{24}{120}, \quad \frac{1}{8} = \frac{15}{120}
\]

Adding these fractions together:

\[
\frac{1}{R_t} = \frac{20}{120} + \frac{24}{120} + \frac{15}{120} = \frac{59}{120}
\]

Now, take the reciprocal to find \( R_t \):

\[
R_t = \frac{120}{59} \approx 2.03 \, \text{ohms}
\]

Now that we have the total resistance, we will find the current through the 8-ohm resistor. Using Ohm's Law \( I = \frac{V}{R} \):

The voltage across the parallel resistors is 30 V, and the current through the 8-ohm resistor \( I_3 \) is calculated as:

\[
I_3 = \frac{V}{R_3} = \frac{30}{8} = 3.75 \, \text{A}
\]

Thus, the total resistance of the circuit is approximately 2.03 ohms, and the current through the 8-ohm resistor is 3.75 A.

**Final answers**:
- Total Resistance = 2.03 ohms.
- Current = 3.75 A.

Based on the options provided, the answer would be:

**Total Resistance = 2.03 ohms. Current = 3.75 A.**