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 $$25$ per gallon. Goop purchases raw material for $$10$ per gallon and Goop must spend $$5$ per gallon to dispose of 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.

Suppose Goop purchases $150$ gallons of raw material. What is the probability that they will not run out of raw material? $($Enter the answer as a percentage$)$

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

Suppose Goop purchases $400$ gallons of raw material. How much leftover inventory should they expect to have?

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 raw material should they purchase?

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