If two resistors with resistances R1 and R2 are connected in parallel, as in the figure below, 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 0.3 Ω/s and 0.2 Ω/s, respectively, how fast is R changing when R1 = 80 Ω and R2 = 110 Ω?

0.1360 OHMS PER SECONDS

0.154ohms/s

To find how fast the total resistance R is changing, we will take the derivative of the equation 1/R = 1/R1 + 1/R2 with respect to time.

Let's differentiate both sides of the equation with respect to time (t):

d/dt (1/R) = d/dt (1/R1) + d/dt (1/R2)

To find d/dt (1/R), we use the chain rule:

d/dt (1/R) = -1/R^2 * dR/dt

Given that dR1/dt = 0.3 Ω/s and dR2/dt = 0.2 Ω/s, let's substitute these values into the equation:

-1/R^2 * dR/dt = -1/R1^2 * dR1/dt - 1/R2^2 * dR2/dt

Next, we substitute the given values R1 = 80 Ω and R2 = 110 Ω into the equation:

-1/R^2 * dR/dt = -1/80^2 * 0.3 - 1/110^2 * 0.2

Simplifying this equation:

-1/R^2 * dR/dt = -0.00046875 - 0.00003552

-1/R^2 * dR/dt = -0.00050427

Now, let's solve for dR/dt by multiplying both sides by -R^2:

dR/dt = R^2 * 0.00050427

Finally, substitute the given values R1 = 80 Ω and R2 = 110 Ω back into the equation:

dR/dt = (80^2 + 110^2) * 0.00050427

Solving this equation will give you the rate at which R is changing when R1 = 80 Ω and R2 = 110 Ω.

To find how fast the total resistance R is changing, we need to use the concept of related rates. We have the equation:

1/R = 1/R1 + 1/R2

To find how fast R is changing (dR/dt) when R1 = 80 Ω and R2 = 110 Ω, we can differentiate both sides of the equation with respect to time (t):

d/dt (1/R) = d/dt (1/R1) + d/dt (1/R2)

To simplify, we can first find the derivative of 1/R with respect to R. We have:

d/dR (1/R) = -1/R^2

Next, we need to find the derivatives of 1/R1 and 1/R2 with respect to time:

d/dt (1/R1) = dR1/dt / R1^2
d/dt (1/R2) = dR2/dt / R2^2

Now, let's substitute these derivatives back into our equation:

-1/R^2 = dR1/dt / R1^2 + dR2/dt / R2^2

We can rearrange this equation to solve for dR/dt, which represents how fast R is changing:

dR/dt = -1 / (1/R^2) * (dR1/dt / R1^2 + dR2/dt / R2^2)

Since we are given the values of dR1/dt, dR2/dt, R1, and R2, we can substitute these into the equation and calculate dR/dt.