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An artisan makes hand-crafted painted benches to sell at a craft mall. Her weekly revenue...

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Question

Accounting

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. $)$

$R (n) = - 10.857 n^{2} + 91.714 n,$ 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.

$P^{'} (n) =$

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. $)$

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

per bench

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

by $

$|$ per bench.

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

per bench

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

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

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

per bench

When the artisan makes $10$ benches, the profit is $- - -$ Select $- - v v$ 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.

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