A radio station is giving away tickets to a play. They plan to give away tickets for seats that cost $10 and $20. They want to give away at least 20 tickets. The total cost of all the tickets they give away can be no more than $280.
The Correct Answer and Explanation is:
To solve the problem, we need to set up a system of inequalities based on the given conditions.
Let:
- ( x ) = number of $10 tickets
- ( y ) = number of $20 tickets
Given Conditions:
- The total number of tickets should be at least 20:
[
x + y \geq 20
] - The total cost of the tickets should not exceed $280:
[
10x + 20y \leq 280
] - Both ( x ) and ( y ) must be non-negative:
[
x \geq 0, \quad y \geq 0
]
Simplifying the Cost Inequality
We can simplify the second inequality:
[
10x + 20y \leq 280
]
Dividing the entire inequality by 10 gives:
[
x + 2y \leq 28
]
Summary of Inequalities
We now have the following system of inequalities:
- ( x + y \geq 20 ) (at least 20 tickets)
- ( x + 2y \leq 28 ) (cost constraint)
- ( x \geq 0 )
- ( y \geq 0 )
Finding the Feasible Region
To find the feasible solutions, we can graph these inequalities. Here’s how we approach it:
- Graph the line ( x + y = 20 ):
- When ( x = 0 ), ( y = 20 ).
- When ( y = 0 ), ( x = 20 ).
- Graph the line ( x + 2y = 28 ):
- When ( x = 0 ), ( y = 14 ).
- When ( y = 0 ), ( x = 28 ).
Finding Intersection Points
To find the intersection of the lines ( x + y = 20 ) and ( x + 2y = 28 ), we can solve the equations simultaneously.
From ( x + y = 20 ):
[
y = 20 – x
]
Substituting into ( x + 2y = 28 ):
[
x + 2(20 – x) = 28 \
x + 40 – 2x = 28 \
-x + 40 = 28 \
-x = -12 \
x = 12
]
Substituting ( x = 12 ) back into ( y = 20 – x ):
[
y = 20 – 12 = 8
]
Feasible Solution
The intersection point ( (12, 8) ) satisfies both inequalities.
Conclusion
The radio station can give away 12 tickets priced at $10 each and 8 tickets priced at $20 each, fulfilling both conditions of giving away at least 20 tickets while keeping the total cost under $280. This demonstrates the effective use of inequalities to model real-world scenarios, providing a systematic approach to decision-making in such situations.