A deep-sea research submersible has a maximum payload capacity of 1200 kg1200\text{ kg}1200 kg.
The weights of Type A scientific crates are normally distributed with mean 110 kg110\text{ kg}110 kg and standard deviation 15 kg15\text{ kg}15 kg.
The weights of Type B scientific crates are normally distributed with mean 85 kg85\text{ kg}85 kg and standard deviation 10 kg10\text{ kg}10 kg.
Assume that the weights of individual crates are independent.
Find the probability that when 6 Type A6\text{ Type A}6 Type A crates and 5 Type B5\text{ Type B}5 Type B crates are loaded onto the submersible, the total payload exceeds 1200 kg1200\text{ kg}1200 kg.
The mission commander decides to set a limit on the total number of crates, ccc, that can be carried on a single dive. Find the largest integer value of ccc such that the probability of the total payload exceeding 1200 kg1200\text{ kg}1200 kg is less than 2%2\%2%, regardless of the combination of crate types used.
