A precise steel tape measure has been calibrated at `20^(@)C`. At `40^(@)C` a. will it read high or low, and b. what will be the percentage error?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
The steel tape measure is calibrated at 20°C and is used at 40°C. We need to determine whether it will read high or low and calculate the percentage error in the measurement.

### Step 2: Determine the effect of temperature on the steel tape
Steel expands when heated. Since the tape is calibrated at 20°C, its markings are accurate at that temperature. When the temperature increases to 40°C, the tape itself will expand, causing the markings to shift outward. This means that a measurement taken with the tape at 40°C will be less than the actual length because the tape's markings have expanded.

### Step 3: Answer the first part of the question
Since the tape expands and the markings move outward, the tape will read low at 40°C.

**Answer (a)**: The tape will read low.

### Step 4: Calculate the percentage error
To find the percentage error, we will use the formula:

\[ \text{Percentage Error} = \frac{\text{Actual Length} - \text{Measured Length}}{\text{Actual Length}} \times 100 \]

Let \( L_0 \) be the actual length of the object, and let \( MV \) be the measured value (length measured with the tape at 40°C).

### Step 5: Relate the actual length to the measured length
The relationship between the actual length and the measured length can be expressed as:

\[ L_0 = MV \times (1 + \alpha_s \Delta T) \]

Where:
- \( \alpha_s \) is the coefficient of linear expansion for steel (approximately \( 12 \times 10^{-6} \, \text{°C}^{-1} \))
- \( \Delta T = 40°C - 20°C = 20°C \)

### Step 6: Solve for the measured value
Rearranging the equation gives:

\[ MV = \frac{L_0}{1 + \alpha_s \Delta T} \]

### Step 7: Substitute into the percentage error formula
Now substituting \( MV \) into the percentage error formula:

\[ \text{Percentage Error} = \frac{L_0 - \frac{L_0}{1 + \alpha_s \Delta T}}{L_0} \times 100 \]

This simplifies to:

\[ \text{Percentage Error} = \left(1 - \frac{1}{1 + \alpha_s \Delta T}\right) \times 100 \]

### Step 8: Calculate the percentage error
Substituting \( \alpha_s = 12 \times 10^{-6} \) and \( \Delta T = 20 \):

\[ \text{Percentage Error} = \left(1 - \frac{1}{1 + (12 \times 10^{-6} \times 20)}\right) \times 100 \]

Calculating \( \alpha_s \Delta T = 12 \times 10^{-6} \times 20 = 0.00024 \):

\[ \text{Percentage Error} = \left(1 - \frac{1}{1 + 0.00024}\right) \times 100 \]

Using the binomial approximation for small values:

\[ \text{Percentage Error} \approx 0.00024 \times 100 = 0.024\% \]

### Final Answer
**Answer (b)**: The percentage error is approximately 0.024%.

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