Let's solve each part step by step:

Part (a) – Find the scale interval of the ruler

The percentage error of a measurement can be expressed as:

$\text{Percentage Error} = \frac{\text{Absolute Error}}{\text{Measured Value}} \times 100$

The absolute error is the smallest possible measurement that can cause a deviation. Since the percentage error is given as 10%, we can rearrange the formula to find the absolute error.

$\text{Absolute Error} = \frac{\text{Percentage Error} \times \text{Measured Value}}{100}$

Given:

• Measured Value of pencil = 10 cm
• Percentage Error = 10%

Substitute the values:

$\text{Absolute Error} = \frac{10 \times 10}{100} = 1 \, \text{cm}$

This means that the absolute error is 1 cm. The absolute error represents the scale interval of the ruler, because the smallest unit the ruler can measure with this error is 1 cm.

Thus, the scale interval of the ruler is 1 cm.

Part (b) – Find the relative error for a 12 cm measurement

To find the relative error, we first calculate the absolute error based on the scale interval, then divide it by the new measured value (12 cm).

Given:

• The absolute error from part (a) is 1 cm.
• Measured Value = 12 cm.

The relative error is given by:

$\text{Relative Error} = \frac{\text{Absolute Error}}{\text{Measured Value}}$

Substituting the values:

$\text{Relative Error} = \frac{1}{12} = 0.0833$

To express this as a percentage, multiply by 100:

$\text{Relative Error (percentage)} = 0.0833 \times 100 = 8.33\%$

So, the relative error for the measurement of 12 cm is 8.33%.

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Summary of Answers:

• a) The scale interval of the ruler is 1 cm.
• b) The relative error for the measurement of 12 cm is 8.33%.

Would you like more details or have any other questions? Here are some related questions:

1. How would the relative error change if the ruler had a smaller scale interval?
2. What happens to the absolute error if the measurement is taken more precisely?
3. Can the absolute error in part (a) be smaller than the scale interval? Why or why not?
4. How do you calculate percentage error if you know the true value and the measured value?
5. What is the difference between absolute error and relative error in measurements?

Tip: Always consider the scale interval of the measuring instrument when calculating errors—it directly affects the precision of the measurement.