Question - Optimizing Ticket Sales Revenue
Solution:
The problem describes a scenario where a local theater is selling tickets to a production, with the goal of making at least $15,000 to proceed with the show. The theater has a maximum capacity of 1,000 people. Adult tickets are priced at $25 each and youth tickets at $12.50 each. We will denote the number of adult tickets sold as $$ a $$ and the number of youth tickets sold as $$ y $$.We have two main constraints described by the problem:1. The total number of tickets sold cannot exceed the theater capacity:\[ a + y \leq 1000 \]2. The total revenue from the ticket sales must be at least $15,000:\[ 25a + 12.5y \geq 15000 \]We can now form a system of inequalities that represents these constraints:\[ \left\{ \begin{array}{rcl}a + y & \leq & 1000 \\25a + 12.5y & \geq & 15000\end{array}\right. \]The feasible region represented by these inequalities would be graphed in the first quadrant on a coordinate plane where the x-axis represents the number of adult tickets sold ($$ a $$) and the y-axis represents the number of youth tickets sold ($$ y $$).To determine which of the given points are solutions to the system, we can plug in the values for $$ a $$ and $$ y $$ into both inequalities.Let's check each point:1. $$ (4, 1) $$ - $$ 4 + 1 \leq 1000 $$, true - $$ 25 \cdot 4 + 12.5 \cdot 1 = 100 + 12.5 = 112.5 $$, but we need at least $15,000, false2. $$ (7, 1) $$ - $$ 7 + 1 \leq 1000 $$, true - $$ 25 \cdot 7 + 12.5 \cdot 1 = 175 + 12.5 = 187.5 $$, but we need at least $15,000, false3. $$ (5, 4) $$ - $$ 5 + 4 \leq 1000 $$, true - $$ 25 \cdot 5 + 12.5 \cdot 4 = 125 + 50 = 175 $$, but we need at least $15,000, false4. $$ (8, 3) $$ - $$ 8 + 3 \leq 1000 $$, true - $$ 25 \cdot 8 + 12.5 \cdot 3 = 200 + 37.5 = 237.5 $$, but we need at least $15,000, false5. $$ (1, 12) $$ - $$ 1 + 12 \leq 1000 $$, true - $$ 25 \cdot 1 + 12.5 \cdot 12 = 25 + 150 = 175 $$, but we need at least $15,000, falseNone of the given points meet both criteria, so none are valid solutions for the problem under the constraints given. The feasible region would include points where both inequalities are satisfied, which is not the case for any of the given points. The points provided do not reflect realistic ticket sales that would allow the theater to meet its revenue goal, as they all produce revenue figures that are much less than the minimum requirement of $15,000.
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