To help you with this problem, let's break it down step by step.
Step 1: Write the inequality.
The total cost for the event can be represented by the equation:
\[ \text{Total Cost} = (15 \cdot x) + (20 \cdot y) \]
Here, \(15 \cdot x\) represents the entrance fee for \(x\) people, and \(20 \cdot y\) represents the hourly charge for \(y\) hours. Since the budget for the event is $200, the inequality representing this situation would be:
\[ 15x + 20y \leq 200 \]
Step 2: Calculate the cost for 7 people attending for 3 hours.
For \(x = 7\) and \(y = 3\):
\[ \text{Total Cost} = (15 \cdot 7) + (20 \cdot 3) \] \[ = 105 + 60 \] \[ = 165 \]
So, the cost for 7 people to attend for 3 hours is $165.
Step 3: Check if 10 people for a certain number of hours is within the $200 budget.
To determine if the cost for 10 people will fit within the budget, we need to find the maximum \(y\) (number of hours) such that:
\[ 15(10) + 20y \leq 200 \] \[ 150 + 20y \leq 200 \] \[ 20y \leq 200 - 150 \] \[ 20y \leq 50 \] \[ y \leq \frac{50}{20} \] \[ y \leq 2.5 \]
Therefore, for 10 people, they would be able to bowl for 2.5 hours or less to stay within the budget.
Step 4: Check if 8 people for a certain number of hours is within the $200 budget.
Next, let's check if 8 people can bowl for a certain number of hours while staying under the budget:
\[ 15(8) + 20y \leq 200 \] \[ 120 + 20y \leq 200 \] \[ 20y \leq 200 - 120 \] \[ 20y \leq 80 \] \[ y \leq \frac{80}{20} \] \[ y \leq 4 \]
Thus, for 8 people, they can bowl for up to 4 hours while remaining within the budget.
Summary of Answers:
- The inequality is \(15x + 20y \leq 200\).
- The cost for 7 people for 3 hours is $165.
- For 10 people, they can bowl for up to 2.5 hours (within budget).
- For 8 people, they can bowl for up to 4 hours (within budget).
If you need further clarification or assistance, feel free to ask!
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