An artisan makes hand$-$crafted painted benches to sell at a craft mall. Her weekly revenue and costs $($not including labor$)$ are given in the table.

Weekly Revenue and Costs for Hand$-$crafted Benches

$($a$)$ Find a quadratic model for revenue. $($Round all numerical values to three decimal places.$)$

$R(n)=-10\mathrm{.}857n^2+91\mathrm{.}714n,$ dollars

Find a cubic model for cost. $($Round all numerical values to three decimal places.$)$

$C(n)=$

dollars

Using the previous models for revenue and cost, find a cubic model for profit, $P.($Round all numerical values to three decimal places.$)$

$P(n)=$

dollars

$($b$)$ Write the derivative formula for profit.

$P^{\text{'}}(n)=$

per bench

$($c$)$ Find and interpret the rates of change of profit when the artisan sells $5,9,$ and $10$ benches, respectively. $($Round your answers to two decimal places.$)$

$P^{\text{'}}(5)=q$

per bench

When the artisan makes $5$ benches, the profit is

by $

$|$ per bench.

$P^{\text{'}}(9)=$$

per bench

When the artisan makes $9$ benches, the profit is

$|$by $$[,]$ per bench.

$P^{\text{'}}(10)=$$

per bench

When the artisan makes $10$ benches, the profit is $---$ Select$--vv$ by $ per bench.

$($d$)$ Use the information in part $($c$)$ to determine the number of benches the artisan should produce each week for maximum profit.