when two resistors,R1 and R2 are in series, they have an equivalent resistance of 180 ohms, and when they are in parallel, they have an equivalent resistance of 40 ohms. what are their values?
R1 + R2 = 180
1/R1 + 1/R2 = 1/40
Plug one in the other and solve.
To find the values of the resistors R1 and R2, we can use two equations based on the given information:
1. For resistors in series, the equivalent resistance is the sum of individual resistances:
R_series = R1 + R2 = 180 ohms
2. For resistors in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances:
1 / R_parallel = 1 / R1 + 1 / R2
R_parallel = (R1 * R2) / (R1 + R2) = 40 ohms
Now we have a system of two equations:
Equation 1: R1 + R2 = 180
Equation 2: (R1 * R2) / (R1 + R2) = 40
We can solve this system of equations to find the values of R1 and R2.
Rearrange Equation 2:
(R1 * R2) = 40 * (R1 + R2)
Expand and rearrange again:
R1 * R2 = 40R1 + 40R2
Rearrange once more:
R1 * R2 - 40R1 - 40R2 = 0
Now, we have a quadratic equation in two variables (R1 and R2). We can solve this equation by substituting R2 as x and R1 as y. Rewriting the equation:
xy - 40x - 40y = 0
This equation can be written as:
xy - 40x - 40y + 1600 = 1600
(xy - 40x - 40y + 1600) + 1600 = 3200
(x-40)(y-40) = 3200
Now we need to find two factors of 3200 whose values satisfy the equation, (x-40)(y-40) = 3200.
The factors of 3200 include (1, 3200), (2, 1600), (4, 800), (5, 640), (8, 400), (10, 320), (16, 200), (20, 160), (25, 128), (32, 100), (40, 80), and so on.
Using these factors, we can find the values of R1 and R2 by adding 40 to both factors:
R1 = x + 40
R2 = y + 40
For example, when x = 1 and y = 3200, R1 = 1 + 40 = 41 and R2 = 3200 + 40 = 3240.
Continue using other factor pairs until we find the values of R1 and R2 that satisfy the equations:
R1 + R2 = 180
(R1 * R2) / (R1 + R2) = 40
By solving this system of equations, you can find the values of R1 and R2.