$C(x)=72,000+60x$

$p=200-\frac{x}{30}$

a$)$ Write the Revenue and Profit functions and determine their domain.

b$)$ Sketch a careful graph of the Revenue and Cost functions $($the domain of the functions must be clearly indicated on the graph$).$ On the graphs, indicate where the break$-$even points are and the production levels $($values of $x)$ that result in profit, and the ones that result in loss.

$x$

$2.$ Optimization problem

The average cost per unit when producing $x$ units of a commodity is given by the equation

$\frac{?}{\text{b}\text{a}\text{r}}(C(x))$$$3\mathrm{.}00$$$0\mathrm{.}50$$$0\mathrm{.}50$$$3\mathrm{.}00S(x)=x^3-2x^2+4x-8,0x6\frac{x}{b}ar(C(x))=\frac{0\mathrm{.}25x^2+4x+100}{x},0\frac{?}{b}$

$ar(C(x))in$ dollars.

Find the production level that will minimize the average cost and determine the minimum average cost. Verify that what you found $is$ indeed a minimum. Write your answer $in$ full sentence form specifying the values $of$ all the variables $in$ the problem and including appropriate units.

$3.$ Optimization problem

The Transit Authority $(TA)in$ a metropolitan area operates a subway line that currently carries $an$ average $of6000$ passengers per day commuting between the suburbs and downtown. The fare now $is$ $$3\mathrm{.}00$ per ride and the board $of$ the $TAis$ considering raising its price $by$ $$0\mathrm{.}50to$ generate a larger revenue. A study reveals that for each $$0\mathrm{.}50$ increase $in$ fare, the ridership will $be$ reduced $byan$ average $of1000$ passengers per day. Using this study, the board determines that, $to$ get maximum revenue, the price $of$ the ticket should remain $at$ $$3\mathrm{.}00.Is$ this correct? Explain why $or$ why not. Show ALL your work $to$ support your answer.

$4.$ Point $of$ Diminishing Returns

Assume that $an$ advertising campaign $is$ launched for a new product and that the daily sales $of$ the product $(in$ thousands $of$ dollars$)$ are given $by$ the following equation

$S(x)=x^3-2x^2+4x-8,0x6$

when $x$ thousand dollars $is$ spent $on$ advertising.

$Is$ there a point $of$ diminishing returns $in$ this situation? $Ifso,$ determine what $is$ the amount $of$ money spent $on$ advertising that corresponds $to$ the point $of$ diminishing returns and explain what that means.