To determine the best option and break-even point, we can calculate the costs for both visiting options based on the number of visits.
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Cost for Pay per Visit: \[ \text{Cost} = \text{Price per visit} \times \text{Number of visits} = 28 \times \text{Number of visits} \]
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Cost for Membership: \[ \text{Cost} = \text{Membership fee} + (\text{Price per visit after membership} \times \text{Number of visits}) = 90 + 15 \times \text{Number of visits} \]
Now we can analyze the specific scenarios:
Scenario 1: Planning to attend 3 times
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Pay per visit: \[ 28 \times 3 = 84 \]
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Membership: \[ 90 + 15 \times 3 = 90 + 45 = 135 \]
Best option: Pay per visit (84 < 135)
Scenario 2: Planning to attend 8 times
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Pay per visit: \[ 28 \times 8 = 224 \]
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Membership: \[ 90 + 15 \times 8 = 90 + 120 = 210 \]
Best option: Buy a membership (210 < 224)
Scenario 3: Budgeted $150
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Pay per visit (x visits): \[ 28 \times x \] We want this to be less than or equal to 150.
\[ 28x \leq 150 \Rightarrow x \leq \frac{150}{28} \approx 5.36 \]So, if you go 5 times: \[ 28 \times 5 = 140 \]
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Membership: \[ 90 + 15x \leq 150 \Rightarrow 15x \leq 60 \Rightarrow x \leq 4 \]
Best option: Pay per visit (140 < 150, while membership would only allow 4 visits).
Break-even point
Set the costs equal to find the break-even point: \[ 28x = 90 + 15x \] \[ 28x - 15x = 90 \] \[ 13x = 90 \] \[ x = \frac{90}{13} \approx 6.92 \]
Thus, rounding up, the break-even point is about 7 visits.
Summary of Responses
- You have plans to attend 3 times this summer: Pay per visit
- You think you will visit probably about 8 times this summer: Buy a membership
- You have budgeted $150 for visiting the park this summer: Pay per visit
- How many visits would be approximately the break-even point: 7