Questionb) Two resistors are connected in parallel so that their total resistance $R$ is 3.6 ohn resistor's resistance must be 3 ohms greater than the other; find the other's if the formula for total resistance in parallel is; $R=\frac{1}{r_{1}}+\frac{1}{r_{2}}$.

Studdy Solution
Solve the equation for the unknown resistance.
First, find a common denominator for the right-hand side of the equation:
$$\frac{1}{3.6} = \frac{r_1 + 3 + r_1}{r_1(r_1 + 3)}$$
Simplify the numerator:
$$\frac{1}{3.6} = \frac{2r_1 + 3}{r_1(r_1 + 3)}$$
Cross-multiply to solve for $r_1$:
$$r_1(r_1 + 3) = 3.6(2r_1 + 3)$$
Expand both sides:
$$r_1^2 + 3r_1 = 7.2r_1 + 10.8$$
Rearrange the equation:
$$r_1^2 + 3r_1 - 7.2r_1 - 10.8 = 0$$
Simplify:
$$r_1^2 - 4.2r_1 - 10.8 = 0$$
Solve this quadratic equation using the quadratic formula:
$$r_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
where $a = 1$, $b = -4.2$, and $c = -10.8$.
Calculate the discriminant:
$$b^2 - 4ac = (-4.2)^2 - 4(1)(-10.8)$$ $$= 17.64 + 43.2$$ $$= 60.84$$
Calculate $r_1$:
$$r_1 = \frac{4.2 \pm \sqrt{60.84}}{2}$$
Calculate the square root:
$$\sqrt{60.84} \approx 7.8$$
So:
$$r_1 = \frac{4.2 + 7.8}{2}$$ $$r_1 = \frac{12}{2}$$ $$r_1 = 6$$
Since $r_2 = r_1 + 3$, the resistance of the other resistor is:
$$r_2 = 6 + 3 = 9$$
The resistance of the other resistor is:
$\boxed{6}$