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.
r2 = 3r1
r2 = 2r3
Substitute and solve for r2 and evaluate for r1 and r3.
Show your work if you get stuck.
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.
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.