All necessary mathematical work must be shown, and all explanations must be given clearly, thoroughly, and using college$-$level writing. This project may be handwritten, typed, or a combination of both. Your final submission must be a single pdf document submitted into the assignment on Canvas. A company has a cost function for a product of C$($x$)=4$x$0\mathrm{.}02$x$2350.$ The supply and demand functions are given by$,$ respectively, S$($x$)=0\mathrm{.}05$x$28$ and D$($x$)=481\mathrm{.}25$x$.$ Answer the following questions about this company.

a$.)$ Find the equilibrium price and quantity sold for this product.

b$.)$ Construct the graph of supply and

demand, labeling the equilibrium point.

c$.)$ Determine the revenue function for the product if it sells at the equilibrium price.

d$.)$ Graph the cost and revenue function

e$.)$ What is the break$-$even point? $($i$.$e$.$ How many items do they need to sell to become profitable?$)$ Label this point on your cost$/$revenue graph.

f$.)$ Construct the profit function given the revenue function you created in part $($c$.).$

g$.)$ How much profit is the company making at the equilibrium point?

h$.)$ Determine the marginal profit at equilibrium, and explain what this means.

i$.)$ Determine the point elasticity of demand for the product at equilibrium. Explain what this number tells us$.$

j$.)$ Considering the result from part $($i$),$ in order to maximize revenue, should the company raise or lower the price? Or should they keep it the same? Explain your reasoning.

k$.)$ Find the consumers surplus and the producers surplus at equilibrium. Explain what these mean. Label them on your supply$/$demand graph.