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.

a$.$ Suppose Good purchases $150$ gallons of raw material. What is the probability that they will run out of raw material?

b$.$ Suppose Goop purchases $300$ gallons of raw material. How many gallons of demand on average would remain unfulfilled?

c$.$ Suppose Goop purchases $400$ gallons of raw material. How much should they expect to spend on disposal costs $($in $s$)?$

d$.$ Suppose Goop wants to ensure that there is a $92\%$ probability that they will be able to satisfy the customer$$s entire demand. How many gallons of the raw material should they purchase?

e$.$ How many gallons should Goop purchase to maximize its expected profit?