Question

If three resistors with resistance of R1, R2, R3 are wired in parralle, their combined combined resitance R is given by the following formula:

1/R= 1/R1 + 1/R2 +1/R3

The design of a voltmeter requires that the resitance R2 be 80 ohms greater than the restance of R1, that the restance of R3 be 5 ohms greater than R1, and that their combined restance be 10 ohms. Find the value of each resistance.

R1=________ohms

R2=________ohms

R3=________ohms

Answer 1

If three resistors with resistance of R₁, R₂, R₃ are wired in parallel, their combined resistance R is given by the following formula:

1/R= 1/R₁ + 1/R₂ +1/R₃.

Since the resistance of R₂ is 80 ohms greater than the resistance of R1, hence R₂ = R₁+80…(1). Also, since the resistance of R₃ is 5 ohms greater than the resistance of R₁, hence R₃= R₁+5…(2). Further, since the combined resistance of R₁,R₂ and R₃ is 10 ohms, hence R = 10 so that 1/10 = (1/R1) +(1/R2) + (1/R3) or, 1/10 = (1/R₁) + [1/(R₁+80)]+ [1/( R₁+5)] or, 1/10 = (3R₁²+170R₁+400)/[R₁(R₁+5)(R₁+80)] or, R₁(R₁+5)(R₁+80) = 10(3R₁²+170R₁+400) or, (R₁)³+55(R₁)²-1300R₁- 4000 = 0 or, (R₁-20) (R₁² +75R₁+200) = 0

A graph of (R₁)²+75R₁+200 is attached. It may be observed that this graph crosses the X-Axis/R₁ at R₁ =              -72.231 and R₁ = -2.769. Further, since R₁ cannot be negative, hence R₁ = 20 ohms. Then R₂ = 20+80 = 100 ohms and R₃ = 20+5 = 25 ohms.

Note:

1. We can use the quadratic formula also on the equation (R₁)²+75R₁+200 to arrive at R₁= -72.231 or,            R₁ = -2.769.

2. The exact coordinates may be seen only in the original Desmos graph and not in an image thereof.