$[$The following information applies to questions $1-5.]$

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.

$1)$ 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$)$

$2)$ Suppose Goop purchases $300$ gallons of raw material. How many gallons of demand on average would remain unfulfilled?

$3)$ Suppose Goop purchases $400$ gallons of raw material. How much leftover inventory should they expect to have?

$4)$ 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?

$5)$ How many gallons should Goop purchase to maximize its expected profit?