Part of a circuit has two resistors connected in parallel as shown in the figure. R(1) is a constant resistor of 10 ohm, while R(2) is a variable resistor with resistance that varies at a rate of 2 ohms per minute. The total, or effective, resistance, R, provided by this circuit is given by:
1/R=1/R(1)+1/R(2)
At what rate does the effective resistance change when R(2) is 100 ohms? Round off your answer to four decimal places
R = R1 R2/(R1+R2)
R1 is constant
dR = R1 [(R1+R2) dR2- R2 dR2]/ (R1+R2)^2
dR/dt = R1 dR2/dt [R1]/(R1+R2)^2
dR/dt = 100(2) /(110)^2
0.0165
To find the rate at which the effective resistance changes when R(2) is 100 ohms, we need to differentiate the equation 1/R = 1/R(1) + 1/R(2) with respect to time.
Let's denote the effective resistance as R(t) and the variable resistance as R(2)(t), where t represents time.
Differentiating the equation, we have:
d(1/R)/dt = d(1/R(1))/dt + d(1/R(2))/dt
To find the rate at which the effective resistance changes when R(2) is 100 ohms, we need to substitute R(2) with 100 ohms in the equation and solve for d(1/R)/dt.
Given:
R(1) = 10 ohms
dR(2)/dt = 2 ohms per minute
R(2) = 100 ohms
Let's calculate the rate at which the effective resistance changes:
1/R = 1/R(1) + 1/R(2)
Substituting values:
1/R = 1/10 + 1/100
1/R = 10/100 + 1/100
1/R = 11/100
R = 100/11 ohms
Now, differentiate both sides with respect to time:
d(1/R)/dt = d/dt(11/100)
d(1/R)/dt = 0 (since 11/100 is a constant)
Therefore, the rate at which the effective resistance changes when R(2) is 100 ohms is 0 ohms per minute.
To find the rate at which the effective resistance changes when R(2) is 100 ohms, we need to differentiate the equation with respect to time.
1/R = 1/R(1) + 1/R(2)
Differentiating both sides with respect to time (t), we get:
d(1/R)/dt = d(1/R(1))/dt + d(1/R(2))/dt
The first term, d(1/R)/dt, represents the rate of change of the effective resistance. The second term, d(1/R(1))/dt, is zero because R(1) is a constant resistor. The third term, d(1/R(2))/dt, represents the rate of change of R(2).
Since R(1) is constant, d(1/R(1))/dt = 0.
To find d(1/R(2))/dt, we need to differentiate 1/R(2) with respect to t.
d(1/R(2))/dt = -1/R(2)^2 * dR(2)/dt
We are given that dR(2)/dt = 2 ohms per minute, and we need to find d(1/R)/dt when R(2) = 100 ohms.
Substituting the given values into the equation, we have:
d(1/R)/dt = -1/(100^2) * 2
Simplifying the expression, we get:
d(1/R)/dt = -1/10000 * 2 = -0.0002
Therefore, the effective resistance changes at a rate of -0.0002 ohms per minute when R(2) is 100 ohms.