Sure, let's find the value of the second resistor, [tex]\( R_2 \)[/tex], given the total resistance [tex]\( R_T \)[/tex] and the first resistor [tex]\( R_1 \)[/tex]:
The formula for the total resistance [tex]\( R_T \)[/tex] in a parallel circuit with resistors [tex]\( R_1 \)[/tex] and [tex]\( R_2 \)[/tex] is:
[tex]\[ \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} \][/tex]
We are given the values:
- [tex]\( R_T = 2 \)[/tex] ohms
- [tex]\( R_1 = 6 \)[/tex] ohms
First, we calculate the reciprocal of the total resistance:
[tex]\[ \frac{1}{R_T} = \frac{1}{2} = 0.5 \][/tex]
Next, we find the reciprocal of [tex]\( R_1 \)[/tex]:
[tex]\[ \frac{1}{R_1} = \frac{1}{6} = 0.16666666666666666 \][/tex]
Using these values, we can solve for the reciprocal of [tex]\( R_2 \)[/tex]:
[tex]\[ \frac{1}{R_2} = \frac{1}{R_T} - \frac{1}{R_1} \][/tex]
Substituting the known values:
[tex]\[ \frac{1}{R_2} = 0.5 - 0.16666666666666666 = 0.33333333333333337 \][/tex]
To find [tex]\( R_2 \)[/tex], we take the reciprocal of [tex]\( \frac{1}{R_2} \)[/tex]:
[tex]\[ R_2 = \frac{1}{0.33333333333333337} = 2.9999999999999996 \][/tex]
Therefore, the value of [tex]\( R_2 \)[/tex] is:
[tex]\[ R_2 = 3 \text{ ohms} \][/tex]
