A monopolist operates in a market that can be divided into two distinct segments, each with its own demand function. For the first segment, the demand function is given by $Q_1 = 55 - P_1$, where $Q_1$ is the quantity demanded and $P_1$ is the price in the first segment. For the second segment, the demand function is $Q_2 = 70 - 2P_2$, where $Q_2$ is the quantity demanded and $P_2$ is the price in the second segment. The total cost (TC) for producing the total quantity $Q$, which is the sum of $Q_1$ and $Q_2$, is given by $TC = 5Q$. Assuming the monopolist engages in third-degree price discrimination, Determine the price charged by the monopolist in the second market segment and Calculate the price elasticity of demand facing the firm in the first segment.
$P_2 = \$20$ and $|\epsilon_1| = \frac{6}{5}$
$P_2 = \$20$ and $|\epsilon_1| = \frac{4}{3}$
$P_2 = \$30$ and $|\epsilon_1| = \frac{6}{5}$
$P_2 = \$30$ and $|\epsilon_1| = \frac{4}{3}$
