Resistance of tungsten wire at `150^(@)C` is `133 Omega`. Its resistance temperature coefficient is `0.0045//^(@)C`. The resistance of this wire at `500^(@)C` will be

Options

A. `180 Omega`
B. `225 Omega`
C. `258 Omega`
D. `317 Omega`

Explanation

To find the resistance of the tungsten wire at \(500^\circ C\), we can use the formula for resistance as a function of temperature:

\[ R_T = R_0 \times (1 + \alpha T) \]

Where:
- \(R_T\) is the resistance at temperature \(T\),
- \(R_0\) is the resistance at \(0^\circ C\),
- \(\alpha\) is the temperature coefficient of resistance,
- \(T\) is the temperature in degrees Celsius.

### Step 1: Set up the equation for the known resistance at \(150^\circ C\)

Given:
- Resistance at \(150^\circ C\) (\(R_{150}\)) = \(133 \, \Omega\)
- Temperature coefficient (\(\alpha\)) = \(0.0045 \, ^\circ C^{-1}\)

Using the formula:

\[ R_{150} = R_0 \times (1 + \alpha \times 150) \]

Substituting the known values:

\[ 133 = R_0 \times (1 + 0.0045 \times 150) \]

### Step 2: Calculate the term inside the parentheses

Calculate \(0.0045 \times 150\):

\[ 0.0045 \times 150 = 0.675 \]

Now substitute this back into the equation:

\[ 133 = R_0 \times (1 + 0.675) \]

This simplifies to:

\[ 133 = R_0 \times 1.675 \]

### Step 3: Solve for \(R_0\)

Now, we can solve for \(R_0\):

\[ R_0 = \frac{133}{1.675} \]

Calculating this gives:

\[ R_0 \approx 79.6 \, \Omega \]

### Step 4: Calculate the resistance at \(500^\circ C\)

Now we can use the value of \(R_0\) to find the resistance at \(500^\circ C\):

\[ R_{500} = R_0 \times (1 + \alpha \times 500) \]

Substituting \(R_0\) and \(\alpha\):

\[ R_{500} = 79.6 \times (1 + 0.0045 \times 500) \]

Calculating \(0.0045 \times 500\):

\[ 0.0045 \times 500 = 2.25 \]

Now substitute this back into the equation:

\[ R_{500} = 79.6 \times (1 + 2.25) = 79.6 \times 3.25 \]

Calculating this gives:

\[ R_{500} \approx 258.7 \, \Omega \]

### Final Answer

The resistance of the tungsten wire at \(500^\circ C\) is approximately \(258 \, \Omega\).

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