A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of $187 \mathrm{lb}$ and a standard deviation of $35 \mathrm{lb}$. The water taxi has a stated capacity of 25 passengers, and the water taxi was rated for a load limit of $3500 \mathrm{lb}$. Complete parts (a) through (d) below. a. Given that the water taxi was rated for a load limit of $3500 \mathrm{lb}$, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers? The maximum mean weight is $140 \mathrm{lb}$. (Type an integer or a decimal. Do not round.) b. If the water taxi is filled with 25 randomly selected passengers, what is the probability that their mean weight exceeds the value from part (a)? The probability is (Round to four decimal places as needed.)

Step 1 ：For part a, we need to find the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers. This can be calculated by dividing the total weight limit by the number of passengers. The calculation is \(\frac{3500}{25} = 140\) lb. So, the maximum mean weight is \(\boxed{140}\) lb.

Step 2 ：For part b, we need to find the probability that the mean weight of 25 randomly selected passengers exceeds the value from part a. This is a problem of normal distribution. We can use the z-score formula to calculate the probability. The z-score is the number of standard deviations a particular data point is from the mean. In this case, we need to find the z-score for the mean weight from part a, and then find the probability that a randomly selected data point (passenger's weight) is greater than this z-score. The z-score is calculated as \(\frac{140 - 187}{\frac{35}{\sqrt{25}}} = -6.714285714285714\).

Step 3 ：Using the z-score, we can find the probability that a randomly selected passenger's weight is greater than the z-score. This probability is approximately 1.0000.

Step 4 ：Final Answer: The maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers is \(\boxed{140}\) lb. The probability that the mean weight of 25 randomly selected passengers exceeds this value is approximately \(\boxed{1.0000}\).