Assuming a demand function given by $P=114-2Q$ where $Q$ is the monthly output of a firm in $(000,$ units$)$ and $P$ is price. Let the fixed costs be $36,$ and variable costs be $4.$

$($a$)$ Obtain an expression for profit in terms of $Q$

$(4$ marks$)$

$($b$)$ Find the level of output which gives a profit of $4500-(4$ marks$)$

$($c$)$ Find the maximum profit and value of $P$ and $Q$ if it was achieved

$(12$ marks$)$

$($d$)$ Find the price$-$elasticity of demand at the price and quantity level that gives the maximum output. Comment on the result. marks$)$

$(4$