Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 7 -in and a standard deviation of 1 -in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest $2.9 \%$ or largest $2.9 \%$.

What is the minimum head breadth that will fit the clientele?
\[
\min =
\]

What is the maximum head breadth that will fit the clientele?
\[
\max =
\]

Enter your answer as a number accurate to 1 decimal place.

Step 1 ：First, we need to find the z-scores that correspond to the smallest 2.9% and largest 2.9% of the distribution. In a normal distribution, about 95% of the data falls within two standard deviations of the mean. Therefore, the smallest 2.9% and largest 2.9% correspond to the z-scores that are 2 standard deviations away from the mean.

Step 2 ：We can use a z-table or a calculator to find these z-scores. The z-score for the smallest 2.9% is approximately -1.88, and the z-score for the largest 2.9% is approximately 1.88.

Step 3 ：Now, we can use these z-scores to find the corresponding head breadths. The formula to convert a z-score to an actual value is: \(X = \mu + Z\sigma\), where \(X\) is the value we're trying to find, \(\mu\) is the mean, \(Z\) is the z-score, and \(\sigma\) is the standard deviation.

Step 4 ：For the minimum head breadth: \(\text{min} = \mu + Z\sigma = 7 - 1.88*1 = 5.12 \text{ inches}\)

Step 5 ：For the maximum head breadth: \(\text{max} = \mu + Z\sigma = 7 + 1.88*1 = 8.88 \text{ inches}\)

Step 6 ：So, the helmets will be designed to fit men with head breadths between 5.1 inches and 8.9 inches, to one decimal place. The final answer is \(\boxed{[5.1, 8.9]}\)