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Let's go through both problems step by step.

### 1st Problem (Resistors in Parallel):

The formula for total resistance \( R \) in parallel is given as:

\[
\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}
\]

We need to determine how fast \( R \) is changing, i.e., find \( \frac{dR}{dt} \), given that:
- \( R_1 = 60 \, \Omega \), \( R_2 = 70 \, \Omega \),
- \( \frac{dR_1}{dt} = 0.3 \, \Omega/s \), and \( \frac{dR_2}{dt} = 0.2 \, \Omega/s \).

1. Differentiate the equation \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \) implicitly with respect to time \( t \):

\[
-\frac{1}{R^2} \frac{dR}{dt} = -\frac{1}{R_1^2} \frac{dR_1}{dt} - \frac{1}{R_2^2} \frac{dR_2}{dt}
\]

2. Solve for \( \frac{dR}{dt} \):

\[
\frac{dR}{dt} = R^2 \left( \frac{1}{R_1^2} \frac{dR_1}{dt} + \frac{1}{R_2^2} \frac{dR_2}{dt} \right)
\]

Now, calculate \( R \) first using:

\[
\frac{1}{R} = \frac{1}{60} + \frac{1}{70}
\]

This gives:

\[
R \approx 32.308 \, \Omega
\]

Next, plug in the values to find \( \frac{dR}{dt} \):

\[
\frac{dR}{dt} = (32.308)^2 \left( \frac{0.3}{60^2} + \frac{0.2}{70^2} \right)
\]

Let's compute this.

### 2nd Problem (Triangle with Law of Cosines):

The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cos(\theta)
\]

Where:
- \( a = 9 \, m \),
- \( b = 18 \, m \),
- \( \theta = 60^\circ \), and \( \frac{d\theta}{dt} = 2^\circ/\text{min} \).

We want to find how fast the length of the third side \( c \) is changing, i.e., \( \frac{dc}{dt} \).

1. Differentiate the Law of Cosines with respect to time:

\[
2c \frac{dc}{dt} = 2ab \sin(\theta) \frac{d\theta}{dt}
\]

Solve for \( \frac{dc}{dt} \):

\[
\frac{dc}{dt} = \frac{ab \sin(\theta) \frac{d\theta}{dt}}{c}
\]

You can find \( c \) using the Law of Cosines first with \( \theta = 60^\circ \):

\[
c^2 = 9^2 + 18^2 - 2 \cdot 9 \cdot 18 \cdot \cos(60^\circ)
\]

Then substitute the values into the formula to compute \( \frac{dc}{dt} \).

Would you like further assistance with the detailed calculations for both?

If two resistors with resistances R1 and R2 are connected in parallel, then the total resistance R 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 (in Ω/s) is R changing when R1 = 60 Ω and R2 = 70 Ω?

This problem involves determining the rate at which the total resistance of two parallel resistors changes given their individual rates of change. Learn how to apply implicit differentiation to calculate the change in total resistance for R1 = 60 Ω and R2 = 70 Ω.

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