Let’s break down the components of the problem:
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Inequality for Budget: The total cost for the event consists of the entrance fees for \(x\) people and the hourly bowling charge for \(y\) hours. The cost can be expressed as: \[ 15x + 20y \leq 200 \]
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Calculating the Cost for 7 People for 3 Hours: Plugging \(x = 7\) and \(y = 3\) into the cost equation: \[ \text{Total Cost} = 15(7) + 20(3) \] Calculate each part: \[ 15(7) = 105 \quad \text{and} \quad 20(3) = 60 \] Now sum those amounts: \[ 105 + 60 = 165 \] Therefore, the cost for 7 people to attend for 3 hours is $165.
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For 10 People and Hours: To determine if 10 people can attend for an exact duration that is within the budget, let's set \(x = 10\) and calculate a suitable \(y\): \[ 15(10) + 20y \leq 200 \] Simplifying: \[ 150 + 20y \leq 200 \] Subtracting 150 from both sides: \[ 20y \leq 50 \] Dividing both sides by 20: \[ y \leq 2.5 \] This indicates that 10 people can bowl for at most 2.5 hours while staying within the budget.
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For 8 People: Now we set \(x = 8\) and find an appropriate \(y\): \[ 15(8) + 20y \leq 200 \] Simplifying: \[ 120 + 20y \leq 200 \] Subtracting 120 from both sides: \[ 20y \leq 80 \] Dividing both sides by 20: \[ y \leq 4 \] This indicates that 8 people can bowl for up to 4 hours while staying within the budget.
To summarize:
- The inequality is \(15x + 20y \leq 200\).
- The cost for 7 people for 3 hours is $165.
- For 10 people, they can have up to 2.5 hours.
- For 8 people, they can have up to 4 hours.
1 answer