A steel tape is callibrated at `20^@ C`. On a cold day when the temperature is `-15^@ C`, what will be the percentage error in the tape ?

Explanation

To solve the problem of finding the percentage error in the steel tape calibrated at \(20^\circ C\) when the temperature drops to \(-15^\circ C\), we will follow these steps:

### Step 1: Identify the given values
- Calibration temperature (\(T_1\)) = \(20^\circ C\)
- Cold day temperature (\(T_2\)) = \(-15^\circ C\)
- Coefficient of linear expansion for steel (\(\alpha\)) = \(1.2 \times 10^{-5} \, \text{°C}^{-1}\)

### Step 2: Calculate the change in temperature (\(\Delta \theta\))
\[ \Delta \theta = T_2 - T_1 = -15 - 20 = -35 \, \text{°C} \]

### Step 3: Use the formula for linear expansion to find the change in length (\(\Delta L\))
The formula for linear expansion is given by:
\[ \Delta L = L \cdot \alpha \cdot \Delta \theta \]
Since we are interested in the percentage error, we can express the change in length as a percentage of the original length \(L\):
\[ \text{Percentage error} = \frac{\Delta L}{L} \times 100 = \alpha \cdot \Delta \theta \times 100 \]

### Step 4: Substitute the values into the percentage error formula
\[ \text{Percentage error} = \alpha \cdot \Delta \theta \times 100 \]
Substituting the values we have:
\[ \text{Percentage error} = (1.2 \times 10^{-5}) \cdot (-35) \cdot 100 \]

### Step 5: Calculate the percentage error
\[ \text{Percentage error} = (1.2 \times 10^{-5}) \cdot (-35) \cdot 100 = -0.042 \% \]

### Conclusion
The percentage error in the steel tape when the temperature drops to \(-15^\circ C\) is \(-0.042\%\).

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