A radio station is giving away tickets. To 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 the tickets cannot exceed $280. A radio station is giving away tickets. To play. They plan to give away tickets for seats that cost 10$ and 20$. They want to give away at least 20 tickets. The total cost of the tickets cannot exceed $280.
The Correct Answer and Explanation is :
Let’s solve this problem step by step.
Problem Breakdown:
- The radio station is giving away tickets for seats that cost $10 and $20.
- They want to give away at least 20 tickets.
- The total cost of the tickets cannot exceed $280.
We need to figure out how many $10 tickets (let’s call this ( x )) and $20 tickets (let’s call this ( y )) satisfy the conditions:
- The total number of tickets must be at least 20:
[
x + y \geq 20
] - The total cost of the tickets cannot exceed $280:
[
10x + 20y \leq 280
]
Step 1: Simplifying the Cost Condition
We can simplify the cost inequality by dividing it by 10:
[
x + 2y \leq 28
]
Now we have two inequalities:
- ( x + y \geq 20 )
- ( x + 2y \leq 28 )
Step 2: Solve the System of Inequalities
We now need to find values of ( x ) and ( y ) that satisfy both inequalities.
Express ( x ) in terms of ( y ):
From the first inequality ( x + y \geq 20 ), solve for ( x ):
[
x \geq 20 – y
]
Substitute this into the second inequality ( x + 2y \leq 28 ):
[
(20 – y) + 2y \leq 28
]
Simplify:
[
20 + y \leq 28
]
[
y \leq 8
]
So, the value of ( y ) (the number of $20 tickets) can range from 0 to 8.
Step 3: Determine the Number of $10 Tickets
Now, using ( y \leq 8 ), we find the corresponding values for ( x ).
For ( y = 8 ), substitute into ( x + y \geq 20 ):
[
x + 8 \geq 20 \implies x \geq 12
]
Thus, when ( y = 8 ), ( x ) must be at least 12. The total cost will be:
[
10(12) + 20(8) = 120 + 160 = 280
]
This satisfies both conditions.
Conclusion:
The radio station can give away:
- 12 $10 tickets and 8 $20 tickets.
This satisfies the total ticket condition (at least 20 tickets) and the total cost condition (not exceeding $280).