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To solve this problem, let's break down the information and approach step-by-step.

### Given Information
- The diameters of bolts are normally distributed.
- Mean (\(\mu\)) = 0.30 inches
- Standard deviation (\(\sigma\)) = 0.01 inches
- We want to find the percentage of bolts with a diameter greater than 0.32 inches.

### Approach
Since this follows a normal distribution, we can use the Z-score formula to convert the specific diameter value (0.32 inches) into a Z-score, which will then help us determine the probability.

### Step 1: Calculate the Z-score
The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is the value of interest (0.32 inches),
- \( \mu \) is the mean (0.30 inches),
- \( \sigma \) is the standard deviation (0.01 inches).

Substituting in the values:
\[
Z = \frac{0.32 - 0.30}{0.01} = \frac{0.02}{0.01} = 2
\]

### Step 2: Find the Probability
Using the Z-score of 2, we refer to a standard normal distribution table or use a calculator to find the probability that a bolt has a diameter greater than 0.32 inches.

For a Z-score of 2, the cumulative probability to the left is approximately 0.9772. Since we are interested in the probability of a diameter **greater** than 0.32 inches, we need to calculate the complement:
\[
P(X > 0.32) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228
\]

### Step 3: Convert to Percentage
To express this as a percentage:
\[
0.0228 \times 100 = 2.28\%
\]

### Conclusion
Approximately **2.28% of bolts** will have a diameter greater than 0.32 inches.

Would you like further details or have any questions about this solution?

### Further Questions
1. How would the answer change if the standard deviation were larger?
2. What percentage of bolts would have a diameter less than 0.28 inches?
3. How is the Z-score affected by changes in mean or standard deviation?
4. How do you interpret a Z-score in a normal distribution?
5. What would the probability be for a diameter between 0.29 and 0.31 inches?

### Tip
When working with normal distributions, remember that the empirical rule can provide quick estimates: approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?

Learn how to find the probability of bolt diameters exceeding 0.32 inches given a normal distribution with mean 0.30 inches and standard deviation 0.01 inches. This problem uses Z-scores to determine probabilities, suitable for Grades 10-12.

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