The weekly downtime X (in hours) for a certain industrial machine has approximately a gamma distribution with alpha = 3.5 and beta = 1.

The weekly downtime X (in hours) for a certain industrial machine has approximately a gamma distribution with alpha = 3.5 and beta = 1.5. The loss L (in dollars) to the industrial operation as a result of this downtime is given by  L = 30X + 2X^2.

a. Find the expected value and the variance of L. Answer: E(L) = 228.4, V(L) = 26,223.75

b. Find an interval that will contain L on approximately 89% of the weeks that the machine is in use. Answer: ($0, $716.60)