the resistance R measured in ohms of an 80 meter long electric wire of radius x in centimeters is given by the formula r = r(x) = 0.0048/x^2
the radius x is given by x=0.1991 + 0.000003T
where T is the temp in Kelvin. How fast is R changing with respect to T when T = 320 K? Express your answer in scientific notation with the mantissa rounded to three decimal places

To find how fast \( R \) is changing with respect to \( T \), we need to compute the derivative of \( R \) with respect to \( T \). This involves applying the chain rule.

We are given two relationships:

1. \( R(x) = \frac{0.0048}{x^2} \)
2. \( x(T) = 0.1991 + 0.000003T \)

To find \( \frac{dR}{dT} \), we use the chain rule:
\[
\frac{dR}{dT} = \frac{dR}{dx} \cdot \frac{dx}{dT}
\]

### Step 1: Find \( \frac{dR}{dx} \)

The formula for \( R \) is \( R(x) = \frac{0.0048}{x^2} \). Differentiate this with respect to \( x \):
\[
\frac{dR}{dx} = -\frac{2 \times 0.0048}{x^3} = -\frac{0.0096}{x^3}
\]

### Step 2: Find \( \frac{dx}{dT} \)

We are given that \( x(T) = 0.1991 + 0.000003T \), so the derivative of \( x \) with respect to \( T \) is:
\[
\frac{dx}{dT} = 0.000003
\]

### Step 3: Apply the chain rule

Now, apply the chain rule:
\[
\frac{dR}{dT} = \frac{dR}{dx} \cdot \frac{dx}{dT} = -\frac{0.0096}{x^3} \cdot 0.000003
\]

### Step 4: Evaluate at \( T = 320 \)

First, find \( x \) at \( T = 320 \):
\[
x(320) = 0.1991 + 0.000003 \times 320 = 0.1991 + 0.00096 = 0.20006
\]

Now substitute \( x = 0.20006 \) into the expression for \( \frac{dR}{dT} \):
\[
\frac{dR}{dT} = -\frac{0.0096}{(0.20006)^3} \times 0.000003
\]

### Step 5: Simplify

First, calculate \( (0.20006)^3 \):
\[
(0.20006)^3 \approx 0.0080024
\]

Now substitute this value into the expression:
\[
\frac{dR}{dT} = -\frac{0.0096

The resistance R measured in ohms of an 80 meter long electric wire of radius x in centimeters is given by the formula r = r(x) = 0.0048/x^2. The radius x is given by x=0.1991 + 0.000003T where T is the temperature in Kelvin. How fast is R changing with respect to T when T = 320 K? Express your answer in scientific notation with the mantissa rounded to three decimal places.

Learn how to determine the rate at which resistance R changes with respect to temperature T for an electric wire using calculus. The problem involves the formula R(x) = 0.0048/x^2 and x(T) = 0.1991 + 0.000003T, with a detailed solution applying the chain rule. Ideal for students in Grades 11-12.

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