If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in Ohms (Ω), is given by:
1/R=1/R1+1/R2
If R1 and R2 are increasing at rates of .3Ω/s and .2Ω/s, respectively, how fast is R increasing when R1=80Ω and R2=100Ω?
To find how fast R is increasing when R1=80Ω and R2=100Ω, we can differentiate both sides of the equation 1/R=1/R1+1/R2 with respect to time.
Let's denote the rate at which R is changing with respect to time as dR/dt, the rate at which R1 is changing with respect to time as dR1/dt, and the rate at which R2 is changing with respect to time as dR2/dt.
Differentiating both sides of the equation gives:
d(1/R)/dt = d(1/R1)/dt + d(1/R2)/dt
Using the property of differentiation that d(1/x)/dt = - (1/x^2)(dx/dt), we can rewrite the equation as:
- (dR/dt)/R^2 = - (dR1/dt)/R1^2 - (dR2/dt)/R2^2
Rearranging the equation to solve for dR/dt gives us:
dR/dt = -R^2 * [(dR1/dt)/R1^2 + (dR2/dt)/R2^2]
Now we can substitute the given values into the equation: R1=80Ω, R2=100Ω, dR1/dt=0.3Ω/s, and dR2/dt=0.2Ω/s.
dR/dt = - (80Ω * 100Ω) * [(0.3Ω/s)/(80Ω)^2 + (0.2Ω/s)/(100Ω)^2]
Simplifying, we get:
dR/dt = - 8000Ω^2 * (0.003Ω/s + 0.0002Ω/s)
dR/dt = - 8000Ω^2 * 0.0032Ω/s
dR/dt ≈ - 25.6Ω^3/s
Therefore, R is decreasing at a rate of approximately 25.6Ω^3/s when R1=80Ω and R2=100Ω.
To find how fast the total resistance R is increasing, we can differentiate the equation with respect to time t. Let's consider R as a function of time R(t), R1 as a function of time R1(t), and R2 as a function of time R2(t).
Given:
1/R = 1/R1 + 1/R2
Differentiating both sides with respect to time t:
d(1/R)/dt = d(1/R1)/dt + d(1/R2)/dt
Using the chain rule:
-1/R^2 * dR/dt = -1/R1^2 * dR1/dt - 1/R2^2 * dR2/dt
Now, we can substitute the values given: dR1/dt = 0.3 Ω/s and dR2/dt = 0.2 Ω/s.
At the specific point in time when R1 = 80 Ω and R2 = 100 Ω, we need to find dR/dt.
Plugging in the values:
-1/R^2 * dR/dt = -1/(80^2) * 0.3 - 1/(100^2) * 0.2
Simplifying:
-1/R^2 * dR/dt = -0.00046875 - 0.0002
-1/R^2 * dR/dt = -0.00066875
Multiplying both sides by -R^2:
dR/dt = (-0.00066875) * (-R^2)
Since we are interested in the rate at which R is increasing, we take the absolute value:
dR/dt = 0.00066875 * R^2
Now, substitute the values at the specific point: R1 = 80 Ω and R2 = 100 Ω:
R = R1 || R2 (parallel resistances formula)
R = (R1 * R2) / (R1 + R2)
R = (80 * 100) / (80 + 100) = 4000 / 180 = 22.222 Ω
dR/dt = 0.00066875 * (22.222)^2
Now, calculate dR/dt to find how fast R is increasing.