Kosty Koffie is a coffee shop in Berkeley, California. The coffee market in Berkeley has two very different types of customers. There are many wealthy working professionals an a large number of considerably less wealthy college students. The demand functions for coffee from these two groups are, respectively:
$$q_P=700-100P_P$$
and
$$q_S=200-40P_S$$
where $q_P$ is the number of coffee drinks demanded by professionals and $q_S$ is the number of coffee drinks demanded by students. $P_P$ is the price of a coffee drink for a professional, and $P_S$ is the price of a coffee drink for a student. Solving the demand functions for the price, $P$, as a function of the quantity demanded, $q$, gives the two inverse demand functions for coffee for these two groups:
$$P_P=7-0.01q_P$$
and
$$P_S=5-0.025q_S$$
The cost of selling $Q$ coffee drinks is:
$$TC(Q)=2Q+100$$
The profit-maximizing quantity of coffee drinks Kosty Koffie will sell to professionals is and the quantity it will sell to students is The price charged by Kosty Koffie for a coffee to a professional will be $\mathrm{\$}$ and the price charged to a student will be $\mathrm{\$}◻$. The amount of economic profit or loss that Kosty Koffie earns is $\mathrm{\$}$.

Step 1 ：Given the demand functions for coffee from professionals and students as $q_P=700-100P_P$ and $q_S=200-40P_S$ respectively, where $q_P$ is the number of coffee drinks demanded by professionals and $q_S$ is the number of coffee drinks demanded by students. $P_P$ is the price of a coffee drink for a professional, and $P_S$ is the price of a coffee drink for a student.

Step 2 ：Solving the demand functions for the price, $P$, as a function of the quantity demanded, $q$, gives the two inverse demand functions for coffee for these two groups: $P_P=7-0.01q_P$ and $P_S=5-0.025q_S$.

Step 3 ：The cost of selling $Q$ coffee drinks is given by the function $TC(Q)=2Q+100$.

Step 4 ：To find the profit-maximizing quantity of coffee drinks Kosty Koffie will sell to professionals, we need to set up the profit function for professionals and then find the quantity that maximizes this function.

Step 5 ：The profit function is given by the difference between total revenue and total cost. Total revenue is the price per coffee drink times the quantity of coffee drinks sold, and total cost is given by the cost function.

Step 6 ：We can substitute the inverse demand function for the price in the total revenue to express it as a function of quantity. Then we can take the derivative of the profit function with respect to quantity and set it equal to zero to find the quantity that maximizes profit.

Step 7 ：Setting up the profit function for professionals: $profi{t}_P=q_P\ast (7-0.01\ast q_P)-2\ast q_P-100$.

Step 8 ：Taking the derivative of the profit function with respect to quantity: $\text{Double subscripts: use braces to clarify}$.

Step 9 ：Setting the derivative equal to zero and solving for $q_P$ gives the profit-maximizing quantity of coffee drinks Kosty Koffie will sell to professionals.

Step 10 ：Final Answer: The profit-maximizing quantity of coffee drinks Kosty Koffie will sell to professionals is $\boxed{{\displaystyle 250}}$.