3 resistor in parallel have an equivalent resistance of 1000 ohms. If R2 is twice the value of R3 and trice the value of R1, what is the value of R1,R2 and R3? Please show me the sol'n. n_n
1/r1 + 1/r2 + 1/r3 = 1/r = 1/1000
r2 = 3r1
r2 = 2r3
Substitute and solve for r2 and evaluate for r1 and r3.
Show your work if you get stuck.
R3 = R Ohms.
R2 = 2R Ohms.
R1 = 2R/3 Ohms.
1/R1 + 1/R2 + 1/R3 = 1/1000
1/(2R/3) + 1/2R + 1/R = 1/1000
3/2R + 1/2R + 1/R = 1/1000
3/2R + 1/2R + 2/2R = 1/1000.
6/2R = 1/1000.
3/R = 1/1000.
R/3 = 1000.
R = 3000 Ohms.
R1 = 2R/3 = 6000/3 = 2000 Ohms.
R2 = 2R = 2*3000 = 6000 Ohms.
R3 = R = 3000 Ohms.
To solve this problem, we need to apply the principles of parallel resistors and work with the given information.
Let's assign variables for the unknown resistor values:
R1 = resistance of the first resistor,
R2 = resistance of the second resistor,
R3 = resistance of the third resistor.
From the given information, we have three equations:
1. The equivalent resistance of the parallel resistors: 1 / R_eq = 1 / R1 + 1 / R2 + 1 / R3 = 1 / 1000.
2. R2 is twice the value of R3: R2 = 2 * R3.
3. R2 is three times the value of R1: R2 = 3 * R1.
Now, let's substitute the value of R2 from equation (3) into equation (1):
1 / 1000 = 1 / R1 + 1 / (3 * R1) + 1 / R3.
Next, we need to find a common denominator for R1 and (3 * R1):
1 / 1000 = (3 / 3R1) + (1 / 3R1) + 1 / R3.
Combining the fractions:
1 / 1000 = (4 / 3R1) + 1 / R3.
To add the fractions, we need a common denominator. The least common multiple of 3R1 and R3 is 3R1R3. So, multiplying each fraction by an appropriate form of 1, we get:
1 / 1000 = (4 * R3 / (3R1 * R3)) + (1 * (3R1) / (3R1 * R3)).
Simplifying:
1 / 1000 = (4R3 + 3R1) / (3R1 * R3).
Now, multiply both sides of the equation by 3R1R3 to eliminate the denominators:
3R1R3 / 1000 = 4R3 + 3R1.
Rearranging the equation:
3R1R3 - 3R1 = 4R3.
Factoring out R1:
R1(3R3 - 3) = 4R3.
Dividing both sides by (3R3 - 3):
R1 = (4R3) / (3R3 - 3).
Now, we can substitute the value of R1 into equation (2) to find R2:
R2 = 3 * R1 = 3 * [(4R3) / (3R3 - 3)].
Simplifying:
R2 = (12R3) / (3R3 - 3).
Now, we can find the values of R1, R2, and R3 by choosing a value for R3. Let's say R3 = 100 ohms.
Using this value, we can calculate:
R1 = (4R3) / (3R3 - 3) = (400) / (297).
R2 = (12R3) / (3R3 - 3) = (1200) / (297).
Therefore, for R3 = 100 ohms, the values of R1, R2, and R3 are approximately:
R1 ≈ 1.35 ohms
R2 ≈ 4.04 ohms
R3 = 100 ohms.
Remember, these values are approximations since we assigned R3 = 100 ohms. By choosing a different value for R3, the results will vary.