A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 44 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 1212 cm with a standard deviation of 0.050.05 cm. For what lengths will a bolt be​ destroyed?

To determine the lengths at which bolts will be destroyed, we need to calculate the range of lengths that are more than 4 standard deviations away from the mean.

### Given:
- Mean length \( \mu = 12.12 \, \text{cm} \)
- Standard deviation \( \sigma = 0.05 \, \text{cm} \)
- Destroy bolts more than 4 standard deviations from the mean.

### Step 1: Calculate the threshold for destruction
The quality-control policy destroys bolts that are more than 4 standard deviations away from the mean. This means bolts with lengths that fall outside the interval:
\[
\mu - 4\sigma \quad \text{and} \quad \mu + 4\sigma
\]
Using the given values:
\[
\mu - 4\sigma = 12.12 - 4(0.05) = 12.12 - 0.20 = 11.92 \, \text{cm}
\]
\[
\mu + 4\sigma = 12.12 + 4(0.05) = 12.12 + 0.20 = 12.32 \, \text{cm}
\]

### Step 2: Conclusion
Bolts with lengths **less than 11.92 cm** or **greater than 12.32 cm** will be destroyed.

Would you like a more detailed explanation or further clarification?

Here are 5 related questions to explore:
1. What is the general formula for determining the range of acceptable values based on standard deviations?
2. How does changing the standard deviation affect the range of bolt lengths to be destroyed?
3. What happens if the tolerance changes to 3 standard deviations instead of 4?
4. How can you apply this concept to quality control in other manufacturing processes?
5. What is the significance of the normal distribution in quality control?

**Tip**: In quality control, larger standard deviations indicate more variability, meaning products are more spread out from the mean.

A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 4 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have a mean length of 12.12 cm with a standard deviation of 0.05 cm. For what lengths will a bolt be​ destroyed?

Find out how to determine the lengths at which bolts are destroyed based on a manufacturer's quality-control policy. With a mean length of 12.12 cm and a standard deviation of 0.05 cm, this problem covers how to use statistics to calculate the range of acceptable lengths and the 4 standard deviations rule.

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