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Chapter 12: Problem 1708

A parallel combination of three resistors takes a current of \(7.5 \mathrm{~A}\) form a \(30 \mathrm{~V}\) supply, It the two resistors are \(10 \Omega\) and $12 \Omega$ find which is the third one? (A) \(4 \Omega\) (B) \(15 \Omega\) (C) \(12 \Omega\) (D) \(22 \Omega\)

Short Answer

Expert verified
The value of the third resistor (R3) is 15 Ω, corresponding to option (B). This is obtained by using the formula for total resistance in a parallel circuit \( \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \) and values for R1, R2 and Rt. After substituting the known values, we get \( \frac{1}{R_3} = \frac{1}{4 Ω} - \frac{1}{10 Ω} - \frac{1}{12 Ω} \). Solving this equation, we find that R3 = 15 Ω.

Step by step solution

01

Write down the formula for total resistance in a parallel circuit

To find the total resistance (Rt) in a parallel circuit with three resistors (R1, R2, and R3), we use the following formula: \[\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\]
02

Write down the known values

We are given the values of the first two resistors R1 and R2, and the total current (I) through the circuit. We also know the voltage (V) across the resistors. The values are: R1 = 10 Ω R2 = 12 Ω I = 7.5 A V = 30 V
03

Find the total resistance using Ohm's Law

Using Ohm's law (V = I * R), we can find the total resistance (Rt) in the circuit. Rt = V / I Plugging in the values: Rt = 30 V / 7.5 A = 4 Ω
04

Substitute the known values in the formula for total resistance

Plug in the values of Rt, R1, and R2 in the formula from Step 1: \[\frac{1}{4 \Omega} = \frac{1}{10 \Omega} + \frac{1}{12 \Omega} + \frac{1}{R_3}\]
05

Solve for the unknown resistor value (R3)

Now, we will solve this equation to find the value of R3: \[\frac{1}{R_3} = \frac{1}{4 \Omega} - \frac{1}{10 \Omega} - \frac{1}{12 \Omega}\] Find the common denominator which is 60, and solve for R3: \[\frac{1}{R_3} = \frac{15 - 6 - 5}{60} = \frac{4}{60}\] Now, flip both the numerator and denominator to find R3: \[R_3 = \frac{60}{4} = 15 \Omega\] The value of the third resistor (R3) is 15 Ω, which corresponds to option (B).

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Most popular questions from this chapter

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