A street vendor constructs the table below on the basis of sales data. $($Note: Consumer expenditure and revenue are terms for the same thing from different perspectives. Consumer expenditure is the amount of money that consumers spend on a product, and revenue is the amount of money that businesses take in by selling the product.$)$

Sales of Roses, Given the Price per Dozen

Price

$($dollars$)$Sales

$($dozen roses$)2016525155301303290$

$($a$)$ Find a quadratic function for the model of the data that gives the quantity sold at x dollars, data from

$20$ x $32.$

$($Round all numerical values to three decimal places.$)$

S$($x$)$

$($b$)$ Find a function for the model for consumer expenditure $($revenue for the vendor$),$ where x is the price in dollars, data from

$20$ x $32.$

$($Round all numerical values to three decimal places.$)$

R$($x$)$

$($c$)$ What price should the street vendor charge to maximize consumer expenditure? $($Round your answer to two decimal places.$)$

$

$($d$)$ If each dozen roses costs the vendor $$12,$ what price should he charge to maximize his profit? $($Round your answer to two decimal places.$)$

$