Two resistors connected in series have an equivalent resistance of 790.6 . When they are connected in parallel, their equivalent resistance is 170.2 . Find the resistance of each resistor.
(small value)
(large value)
To find the resistance of each resistor, let's assign variables to represent these resistors. Let's call the resistance of the smaller resistor R1 and the resistance of the larger resistor R2.
When two resistors are connected in series, their equivalent resistance is the sum of their individual resistances. Therefore, we can write the equation:
R1 + R2 = 790.6 (Equation 1)
When the same two resistors are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances. Mathematically, we can express this as:
1 / (1/R1 + 1/R2) = 170.2 (Equation 2)
Now, we have a system of two equations (Equations 1 and 2) with two unknowns (R1 and R2). We can solve this system of equations to find the values of R1 and R2.
Let's start by rearranging Equation 2 to isolate one of the variables:
1 / (1/R1 + 1/R2) = 170.2
Multiply both sides by (1/R1 + 1/R2):
1 = 170.2 * (1/R1 + 1/R2)
Distribute 170.2 on the right side:
1 = 170.2/R1 + 170.2/R2
Now, we can substitute the value of (R1 + R2) from Equation 1 into this equation:
1 = 170.2/(790.6 - R1) + 170.2/R1
Multiply both sides by R1(790.6 - R1) to eliminate the denominators:
R1(790.6 - R1) = 170.2(R1) + 170.2(790.6 - R1)
Simplify:
790.6R1 - R1^2 = 170.2R1 + 135019.12 - 170.2R1
Combine like terms:
-R1^2 + 790.6R1 = 135019.12
Rearrange to form a quadratic equation:
R1^2 - 790.6R1 + 135019.12 = 0
Now we have a quadratic equation in the form of Ax^2 + Bx + C = 0, where A = 1, B = -790.6, and C = 135019.12.
Using the quadratic formula:
R1 = (-B ± √(B^2 - 4AC)) / (2A)
Substituting the values:
R1 = (-(-790.6) ± √((-790.6)^2 - 4 * 1 * 135019.12)) / (2 * 1)
Simplifying:
R1 = (790.6 ± √(624084.36 - 540076.48)) / 2
R1 = (790.6 ± √84007.88) / 2
Calculating the square root of 84007.88:
√84007.88 = 289.775
Now, let's substitute this value back into the equation to find the possible values of R1:
R1 = (790.6 ± 289.775) / 2
There are two possible values for R1:
R1 = (790.6 + 289.775) / 2 = 540.1875
R1 = (790.6 - 289.775) / 2 = 250.9125
Thus, the resistance of the smaller resistor (R1) can be either 540.1875 or 250.9125.
To find the resistance of the larger resistor (R2), we can subtract the smaller resistor's resistance from the total equivalent resistance:
R2 = 790.6 - R1
For R1 = 540.1875:
R2 = 790.6 - 540.1875 = 250.4125
For R1 = 250.9125:
R2 = 790.6 - 250.9125 = 539.6875
Therefore, the resistance of the smaller resistor is approximately 540.1875 ohms, and the resistance of the larger resistor is approximately 250.4125 ohms.