Points$]$

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An automobile manufacturer sells cars in America and Europe, charging different prices in the two markets. The price function for cars sold in America is $p=39-0\mathrm{.}2x$ thousand dollars $($for $0x195),$ and the price function for cars sold in Europe is $q=9-0\mathrm{.}1y$ thousand dollars $($for $0y90),$ where $x$ and $y$ are the numbers of cars sold per day in America and Europe, respectively. The company's cost function is

$C=19+3(x+y)$ thousand dollars.

$($a$)$ Find the company's profit function. $[$Hint: Profit is revenue from America plus revenue from Europe minus costs, where each revenue is price times quantity.$]$

$P(x,y)=$

$$

$($b$)$ Find how many cars should be sold in each market to maximize profit.

America cars

Europe cars

Also find the price $($in thousand dollars$)$ for each market.

America thousand dollars

Europe thousand dollars