? Question

Rewrite the equation to represent the resistance of resistor [tex]$2, R_2$[/tex], in terms of [tex]$R_T$[/tex] and [tex]$R_1$[/tex].

A. [tex]$R_2 = \frac{R_T - R_1}{R_T R_2}$[/tex]
B. [tex]$R_2 = \frac{R_T - R_3}{R_1 - 1}$[/tex]
C. [tex]$R_2 = \frac{R_T - R_1}{R_1 - R_T}$[/tex]
D. [tex]$R_2 = \frac{R_1 + R_2}{R_T R_2}$[/tex]



Answer :

To rewrite the equation to express the resistance of resistor \(R_2\) in terms of \(R_\tau\) and \(R_1\), we will start with the given equation:
[tex]\[ R_2 = \frac{R_\tau - R_1}{R_\tau R_2} \][/tex]

To isolate \(R_2\), follow these steps:

1. Multiply both sides of the equation by \(R_\tau R_2\) to eliminate the denominator on the right-hand side:
[tex]\[ R_2 \cdot R_\tau R_2 = R_\tau - R_1 \][/tex]

2. This simplifies to:
[tex]\[ R_2^2 \cdot R_\tau = R_\tau - R_1 \][/tex]

3. Divide both sides of the equation by \(R_\tau\) to isolate \(R_2^2\) on the left-hand side:
[tex]\[ R_2^2 = \frac{R_\tau - R_1}{R_\tau} \][/tex]

4. Take the square root of both sides to solve for \(R_2\):
[tex]\[ R_2 = \sqrt{\frac{R_\tau - R_1}{R_\tau}} \][/tex]

Therefore, the resistance of resistor \(R_2\) in terms of \(R_\tau\) and \(R_1\) is:
[tex]\[ R_2 = \sqrt{\frac{R_\tau - R_1}{R_\tau}} \][/tex]