Suppose a product must go through an assembly line made up of five sequential operations. The time it takes to complete each operation is normally distributed with a mean of 180 seconds and standard deviation of 5 seconds. Let X denote the cycle time for the line, so that after X seconds, each operation is supposed to be finished and ready to pass the product to the next operation in the assembly line.

a. If the cycle time X = 180 seconds, what is the probability that all five operations will be completed?

b. What cycle time will ensure that all operations are finished 98% of the time?

c. Suppose that the company wants all operations to be completed within 190 seconds 98% of the time. Further suppose that the standard deviation of an operation can be reduced at a cost of $5,000 per second of reduction (from 5), and any or all operations may be reduced as desired by up to 2.5 seconds. By how much should the standard deviations be reduced to achieve the desired performance level, and how much would that cost?