A hot dog stand is trying to evaluate its cash solvency by projecting their potential income and expenses. There's a flat cost comprising the cost of the cart, licensing, and permit fees totaling to $$4,000.$ Purchased in bulk, each hot dog costs the stand $$0\mathrm{.}75$ to produce and is sold to a customer for $$2.$

$($a$)$ Define functions $R(h)$ and $E(h)$ representing the revenue and expenses, respectively, for each hot dogh sold.

$($b$)$ Define a function $P(h)$ representing the profit of the cart for each hot dog $h$ sold. How many hot dogs must be sold to make a profit?