part 3 Suppose the system's answer to a question is 0.047 with a tolerance of 2%. What is the range of submitted answers that the system will score as correct? Enter your answer as an interval in the form of (a,b). part 4 Suppose the calculations of Part 3 are repeated, only this time rounding to 2 significant digits in each calculation. What will the percent difference be now, as a percentage?

### Part 3: Range of Submitted Answers
Given:
- The system's answer is \( 0.047 \).
- Tolerance is \( 2\% \).

To find the range of acceptable answers, we calculate \( 2\% \) of \( 0.047 \):

\[
\text{Tolerance value} = 0.02 \times 0.047 = 0.00094
\]

Now, calculate the lower and upper bounds:

\[
\text{Lower bound} = 0.047 - 0.00094 = 0.04606
\]
\[
\text{Upper bound} = 0.047 + 0.00094 = 0.04794
\]

Thus, the range of submitted answers that the system will score as correct is:

\[
(0.04606, 0.04794)
\]

### Part 4: Percent Difference after Rounding to 2 Significant Digits
Now, rounding the system's answer to 2 significant digits:

\[
\text{Rounded answer} = 0.047 \text{ rounds to } 0.047 \text{ (since it already has 2 significant digits)}
\]

Thus, there is no change in the answer after rounding, so the percent difference remains \( 0\% \).

However, if the question was asking about rounding before calculating the tolerance range:

1. Round the original tolerance value to 2 significant digits:
   \[
   0.00094 \text{ (already 2 significant digits)}
   \]

Since everything is already to 2 significant digits, there is no change in the final calculation, and hence no percent difference.

Would you like details on any of these steps or further clarification?

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**Relative Questions:**

1. What is the effect of rounding to fewer significant digits on tolerance calculations?
2. How would the answer range change if the tolerance was 5% instead of 2%?
3. What happens to the tolerance interval if the answer was an order of magnitude smaller, say 0.0047?
4. How would you approach a problem where the answer requires a larger tolerance, like 10%?
5. What is the significance of significant digits in scientific calculations?

**Tip:** When dealing with percentages and tolerances, always consider the number of significant digits carefully to ensure precision in your calculations.

Explore how tolerance and percent difference are calculated in math problems, using specific examples and calculations. Learn about the impact of rounding to significant digits on tolerance intervals.

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