Question: The St. Maries plywood plant is part of the Potlatch Corporation's Northwest Division. The plywood superintendent organized a study of the tree diameters that are being shipped to the mill. After collecting a large amount of data on diameters, he concluded that the distribution is approximately normally distributed with a mean of 14.25 inches and a standard deviation of 2.92 inches. Because of the way plywood is made, there is a certain amount of waste on each log because the peeling process leaves a core that is approximately 3 inches thick. For this reason, he feels that any log less than 10 inches in diameter is not profitable for making plywood.

a. Based on the data the superintendent has collected, what is the probability that a log will be unprofitable?

b. An alternative is to peel the log and then sell the core as "peeler logs." These peeler logs are sold as fence posts and for various landscape projects. There is not as much profit in these peeler logs, however. The superintendent has determined that he can make a profit if the peeler log's diameter is not more than 32% of the diameter of the log. Using this additional information, calculate the proportion of logs that will be unprofitable.