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

STEP 1

1. Two resistors are connected in parallel.
2. The total resistance $R$ is 3.6 ohms.
3. One resistor's resistance is 3 ohms greater than the other.
4. The formula for total resistance in parallel is given by $R = \frac{1}{r_1} + \frac{1}{r_2}$ .
5. We need to find the resistance of the other resistor.

STEP 2

1. Define variables for the resistances.
2. Set up the equation using the given formula.
3. Solve the equation for the unknown resistance.

STEP 3

Define variables for the resistances.
Let $r_1$ be the resistance of one resistor, and $r_2$ be the resistance of the other resistor. According to the problem, $r_2 = r_1 + 3$ .

STEP 4

Set up the equation using the given formula for total resistance in parallel.
The formula for total resistance in parallel is:
$\frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2}$
Substitute $R = 3.6$ and $r_2 = r_1 + 3$ into the equation:
$\frac{1}{3.6} = \frac{1}{r_1} + \frac{1}{r_1 + 3}$

STEP 5

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:
$\boxed{6}$

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