During an 8-hour shift, the proportion of time X that a sheet-metal stamping machine is down for maintenance or repairs has a beta distribution with and β. That is,
The cost (in hundreds of dollars) of this downtime, due to lost production and cost of maintenance and repair, is given by
C = 10 + 20X + 4X 2.
a Find the mean and variance of C.
b Find an interval in which C will lie with probability at least 0.75.