$2.$ An amusement park, whose customer set is made up of two markets, adults and children,$$

has developed demand schedules as follows:

Quantity

Price $($$$)$ Adults Children

$51520$

$61418$

$71316$

$81214$

$91112$

$101010$

$1198$

$1286$

$1374$

$1462$

$$The marginal operating cost of each unit of quantity is $$5.($Hint: Because marginal cost is a$$

constant, so is average variable cost. Ignore fixed cost.$)$ The owners of the amusement park$$

want to maximize profits.

a$.$ Calculate the price, quantity, and profit if

$1.$ The amusement park charges a different price in each market.

$2.$ The amusement park charges the same price in the two markets combined.

$3.$ Explain the difference in the profit realized under the two situations.

b$.($Mathematical solution$)$ The demand schedules presented in Problem $2$ can be$$

expressed in equation form as follows $($where subscript A refers to the adult market,$$

subscript C to the market for children, and subscript T to the two markets combined$)$:

Q A $=20-1$PA

Q C $=30-2$PC

Q T $=50-3$PT

$$Solve these equations for the maximum profit that the amusement park will attain$$

when it charges different prices in the two markets and when it charges a single price$$

for the combined market