Problem#$1$: $(30$ points$)$

A small chemical company has estimated that the price and the yearly volume produced of its top$-$selling oxidizing agent are related by the following equation:

$3200-V^2=p$

Where $p$ is the price per gallon of the oxidizing agent and $V$ is the gallons of the agent produced per year. The associated fixed costs are $$4,800$ per year and the variable costs are $$120$ per gallon. Use this information to answer the following:

a$)$ Write out the full equation for the profit made from selling the oxidizing agent in terms of the volume. $(10$ points$)$

b$)$ Determine the optimal volume of the oxidizing agent to be produced per year in gallons and obtain the maximum profit from the optimal volume. Show all of your calculations here. $(10$ points$)+4$

c$)$ The company's small capacity plant that produces the agent normally operates for a product rate of $30$ to $60$ gallons per year. Does the optimal demand from part $($b$)$ fall in this range or not? If no$,$ then will the plant have to reduce its capacity or increase it to maximize its profits? $(10$ points$)$