Goop Inc needs to order a raw material to make a special polymer. The demand for the polymer is forecasted to be Normally distributed with a mean of 250 gallons and a standard deviation of 125 gallons. Goop sells the polymer for $25 per gallon. Goop's purchases raw material for $10 per gallon and Goop must spend $5 per gallon to dispose off all unused raw material due to government regulations. (One gallon of raw material yields one gallon of polymer.) If demand is more than Goop can make, then Goop sells only what they made and the rest of demand is lost.
Show your work for each answer.

a. Suppose Goop purchases 300 gallons of raw material. How many gallons of demand for polymer, in expectation, would remain unfulfilled?

b. Suppose Goop purchases 300 gallons of raw material. What is the probability that they will satisfy all demand for polymer?

c. Suppose Goop wants to ensure an in-stock probability of 92%. How many gallons of the raw material should they purchase?

d. How many gallons should Goop purchase to maximize its expected profit? Calculate the maximal expected profit.