1. A manufacturer is producing metal rods, whose lengths are normally distributed with a mean of $75.0 \mathrm{~cm}$ and a standard deviation of $0.25 \mathrm{~cm}$. If 3000 metal rods are produced, how many will be between $74.5 \mathrm{~cm}$ and $75.5 \mathrm{~cm}$ in length?
Answer
Final Answer: The number of rods that will be between \(74.5 \mathrm{~cm}\) and \(75.5 \mathrm{~cm}\) in length is approximately \(\boxed{2040}\).
Step 1 ：Given that the lengths of the metal rods are normally distributed with a mean of \(75.0 \mathrm{~cm}\) and a standard deviation of \(0.25 \mathrm{~cm}\).
Step 2 ：We are asked to find the number of rods that fall within one standard deviation of the mean, which is between \(74.5 \mathrm{~cm}\) and \(75.5 \mathrm{~cm}\).
Step 3 ：In a normal distribution, about 68% of the data falls within one standard deviation of the mean.
Step 4 ：Therefore, we can calculate the number of rods that fall within this range by multiplying the total number of rods by 0.68.
Step 5 ：Given that the total number of rods produced is 3000, we calculate \(3000 \times 0.68 = 2040\).
Step 6 ：Final Answer: The number of rods that will be between \(74.5 \mathrm{~cm}\) and \(75.5 \mathrm{~cm}\) in length is approximately \(\boxed{2040}\).
