Question

modeling with rational equations
in this activity, you will rearrange and solve a rational equation and find and use the inverse of a rational equation.

as weve seen, for a circuit with two resistors arranged in parallel, we can calculate the total resistance in the circuit, ( r_t ), in ohms, with this equation.
( r_t = \frac{r_1 r_2}{r_1 + r_2} )

question 1
part a

rewrite the equation to represent the resistance of resistor 2, ( r_2 ), in terms of ( r_t ) and ( r_1 ).

( \bigcirc ) ( r_2 = \frac{r_t - r_1}{r_t r_1} )
( \bigcirc ) ( r_2 = \frac{r_t r_1}{r_1 - r_t} )
( \bigcirc ) ( r_2 = \frac{r_1 + r_2}{r_t r_1} )
( \bigcirc ) ( r_2 = \frac{r_t r_1}{r_1 - 1} )

Explanation:

Step1: Eliminate denominator

Multiply both sides by $R_1 + R_2$.
$$R_T(R_1 + R_2) = R_1R_2$$

Step2: Expand left-hand side

Distribute $R_T$ across the sum.
$$R_T R_1 + R_T R_2 = R_1 R_2$$

Step3: Isolate $R_2$ terms

Move $R_T R_2$ to right, $R_1 R_2$ to left.
$$R_T R_1 = R_1 R_2 - R_T R_2$$

Step4: Factor out $R_2$

Extract $R_2$ from right-hand side.
$$R_T R_1 = R_2(R_1 - R_T)$$

Step5: Solve for $R_2$

Divide both sides by $R_1 - R_T$.
$$R_2 = \frac{R_T R_1}{R_1 - R_T}$$

Answer:

$\boldsymbol{R_2 = \frac{R_T R_1}{R_1 - R_T}}$ (second option)