Problem $1$ Consider a market where the demand is given by $D(p)=180-p.$

$($a$)$ Assume for the moment that this market is perfectly competitive. There are $50$ identical firms

operating in this market with cost function $c(y_i)=25y_i^2+81.$ Find

$($i$)$ an individual firm's supply,

$($ii$)$ the industry supply,

$($iii$)$ the market price and the total quantity sold in the market,

$($iv$)$ the individual firm's level of production, and

$($v$)$ the individual firm's profit.