To find the total resistance [tex]\( R \)[/tex] in a parallel circuit with resistances [tex]\( R_1 \)[/tex], [tex]\( R_2 \)[/tex], and [tex]\( R_3 \)[/tex], you use the formula:
[tex]\[ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \][/tex]
Given the resistances:
- The battery has a resistance [tex]\( R_1 = 4 \)[/tex] ohms
- The light bulb has a resistance [tex]\( R_2 = 2 \)[/tex] ohms
- The fan has a resistance [tex]\( R_3 = 3 \)[/tex] ohms
Substitute these values into the formula:
[tex]\[ \frac{1}{R_{\text{total}}} = \frac{1}{4} + \frac{1}{2} + \frac{1}{3} \][/tex]
To find [tex]\( \frac{1}{R_{\text{total}}} \)[/tex]:
1. Calculate [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[ \frac{1}{4} = 0.25 \][/tex]
2. Calculate [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
3. Calculate [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ \frac{1}{3} \approx 0.333 \][/tex]
Now add these values together:
[tex]\[ 0.25 + 0.5 + 0.333 = 1.083333... \][/tex]
Therefore,
[tex]\[ \frac{1}{R_{\text{total}}} \approx 1.083333... \][/tex]
To find [tex]\( R_{\text{total}} \)[/tex], take the reciprocal of [tex]\( 1.083333 \)[/tex]:
[tex]\[ R_{\text{total}} = \frac{1}{1.083333...} \approx 0.923 \text{ ohms} \][/tex]
So, the total resistance in the parallel circuit is approximately [tex]\( 0.923 \)[/tex] ohms. The reciprocal total resistance is expressed as [tex]\( 1.083333 \)[/tex]. The answer consistent with this solution and numerical result:
[tex]\[ \frac{1}{R} = \frac{1}{4} + \frac{1}{2} + \frac{1}{3} \][/tex]
Thus, the correct answer option is:
[tex]\[ \frac{1}{R} = \frac{1}{4} + \frac{1}{2} + \frac{1}{3} \][/tex]
