$7.$ In a given week, a store sells $100$ tablets at $\backslash (\backslash$$ $120\backslash )$ each. A quick market survey indicates that for each $\backslash (\backslash$$ $8\backslash )$ rebate offered to buyers, the number of units sold increases by $32.$

$($a$)$ Assuming a linear demand function $\backslash ($ p$=$p$($x$)\backslash )$ where $\backslash ($ x $\backslash )$ is the number of units sold and $\backslash ($ p$($x$)\backslash )$ is the unit sale price at sales level $\backslash ($ x $\backslash ),$ write the formula for $\backslash ($ p$($x$)\backslash ).$

$($b$)$ Write the formula for the revenue function $\backslash ($ R$=$R$($x$)\backslash ).$

$($c$)$ Assume now that the cost of selling $\backslash ($ x $\backslash )$ tablets consists of a fixed cost $\backslash ($ C$\_\{0\}\backslash )$ of $\backslash (\backslash$$ $3000\backslash )$ and a variable cost based on an acquisition cost of $\backslash (\backslash$$ $45\backslash )$ per tablet. Write the formula for the total cost function $\backslash ($ C$=$C$($x$)\backslash ).$

$($d$)$ Write the formula for the profit function $\backslash ($ P$=$P$($x$)\backslash )$ and graph it on the interval $[35,400].$ What kind of curve do you see? Identify its $\backslash ($ x $\backslash )-$intercepts and both coordinates of its peak. Hint: You can use GeoGebra to graph the function and inspect it

$($e$)$ Determine the marginal profit. Set the marginal profit to zero and solve for $\backslash ($ x $\backslash ).$ Reconcile your answer with $($d$)$ above and explain.

$($f$)$ What is the average profit per unit when the profit is maximized?

$($g$)$ Is the maximum average profit per unit the same as the average profit per unit at maximal profit? Yes or No$?$ Justify your answer.