When Two Resistors Are Joined In Series , The Equivalent Resistance Is 45 Ohm. When The Same Are Connected in Parallel , The eq. Resistance is 10 Ohm. Calculate The Two Resistances.
x+y = 45
1/x + 1/y = 1/10
the two resistors are 15 and 30 ohms
To calculate the resistances of the two resistors, we can use the formulas for the equivalent resistance in series and parallel combinations.
Let's label the resistances as R1 and R2.
When the resistors are connected in series, the equivalent resistance (Rs) is given by:
Rs = R1 + R2
Given that Rs = 45 Ohm, we have:
45 = R1 + R2 ----- Equation 1
When the two resistors are connected in parallel, the equivalent resistance (Rp) is given by:
1/Rp = 1/R1 + 1/R2
Simplifying this equation, we get:
Rp = (R1 * R2) / (R1 + R2)
Given that Rp = 10 Ohm, we have:
10 = (R1 * R2) / (R1 + R2) ----- Equation 2
Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns (R1 and R2). We can solve this system to find the values of R1 and R2.
To simplify the system of equations, we'll solve Equation 1 for R1:
R1 = 45 - R2
Substituting this value of R1 into Equation 2:
10 = ((45 - R2) * R2) / (45 - R2 + R2)
Simplifying further:
10 = (45R2 - R2^2) / 45
Simplifying again:
450 = 45R2 - R2^2
Rearranging the equation:
R2^2 - 45R2 + 450 = 0
This is a quadratic equation, which we can solve for R2 using the quadratic formula:
R2 = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = -45, and c = 450. Substituting these values into the quadratic formula:
R2 = (-(-45) ± √((-45)^2 - 4*1*450)) / (2*1)
Simplifying further:
R2 = (45 ± √(2025 - 1800)) / 2
R2 = (45 ± √(225)) / 2
R2 = (45 ± 15) / 2
This gives us two possible values for R2:
R2 = (45 + 15) / 2 = 30 Ohm
R2 = (45 - 15) / 2 = 15 Ohm
Now we can substitute these values of R2 back into Equation 1 to find the corresponding values of R1:
For R2 = 30 Ohm:
45 = R1 + 30
R1 = 45 - 30
R1 = 15 Ohm
For R2 = 15 Ohm:
45 = R1 + 15
R1 = 45 - 15
R1 = 30 Ohm
Therefore, the two resistances are:
R1 = 15 Ohm
R2 = 30 Ohm