Problem $2.$ An $8-$year Lizzie consumes cherries and apples. Her demand function for cherries is $q_c=m-30p_c+20p_a,$ where $m$ is her income, $p_c$ is price of cherries, and $p_a$ is price of apples. Lizzie's income is $$100,$ and the price of apples is $$1.$

$($a$)$ Are cherries and apples substitutes?

$($b$)$ What is Lizzie's demand function for cherries, when her income is $$100$ and price of applies is $$1.$

$($c$)$ What is Lizzie's inverse demand function for cherries, when her income is $$100$ and price of applies is $$1.$

$($d$)$ At what price would Lizzie buy $30$ cherries?

$($e$)$ Graph Lizzie's inverse demand function.

$($f$)$ Suppose the price of applies rises to $$2\mathrm{.}5.$ Write Lizzie's new inverse demand function and plot it on the previous graph.