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An amusement park that you are excited to try is $28 per visit. You have the option to purchase a monthly membership for $90 and then pay $15 for each visit. (4 points) You have plans to attend 3 times this summer. Which option is best? You think you will visit probably about 8 times this summer. Which option is best? You have budgeted $150 for visiting the park this summer. Which option is best? How many visits would be approximately the break even point where both options would cost about the same?
To determine which option is best in each scenario, let's compare the total cost of each option.
1. For 3 visits:
- Option 1: 3 visits x $28/visit = $84
- Option 2: $90 (membership fee) + 3 visits x $15/visit = $90 + $45 = $135
In this case, Option 1 is the better choice as it costs less.
2. For 8 visits:
- Option 1: 8 visits x $28/visit = $224
- Option 2: $90 (membership fee) + 8 visits x $15/visit = $90 + $120 = $210
In this case, Option 2 is the better choice as it costs less.
3. Budget of $150:
- Option 1: The maximum number of visits within the budget is $150 / $28/visit ≈ 5 visits.
- Option 2: The maximum number of visits within the budget is ($150 - $90) / $15/visit ≈ 4 visits.
In this case, Option 2 is the better choice as it allows for more visits within the given budget.
4. Break-even point:
To find the approximate break-even point, let's determine the number of visits for both options to have a similar cost.
- Option 1: $90 (membership fee) + x visits x $15/visit = $28x
Simplifying, we get x ≈ 5.45 visits.
- Option 2: x visits x $15/visit = $90 + $28x
Simplifying, we get x ≈ 5.45 visits.
Therefore, at approximately 5-6 visits, both options would cost about the same.
In summary, the recommended option depends on the specific scenario:
- For 3 visits: Option 1 is best.
- For 8 visits: Option 2 is best.
- With a budget of $150: Option 2 is best.
- Break-even point: Both options cost about the same at approximately 5-6 visits.
To determine the best option, we need to compare the cost of each option based on the given scenarios. Let's evaluate each scenario step-by-step:
Scenario 1: Planning to attend 3 times this summer
Option 1: Pay per visit
Total cost = $28 per visit × 3 visits
Total cost = $84
Option 2: Purchase monthly membership
Total cost = Cost of membership + (Cost per visit × Number of visits)
Total cost = $90 + ($15 × 3 visits)
Total cost = $90 + $45
Total cost = $135
Based on this scenario, Option 1 (pay per visit) is the best choice as it has a lower total cost of $84 compared to Option 2 which costs $135.
Scenario 2: Planning to attend 8 times this summer
Option 1: Pay per visit
Total cost = $28 per visit × 8 visits
Total cost = $224
Option 2: Purchase monthly membership
Total cost = Cost of membership + (Cost per visit × Number of visits)
Total cost = $90 + ($15 × 8 visits)
Total cost = $90 + $120
Total cost = $210
In this scenario, Option 2 (purchase monthly membership) is the best choice as it has a lower total cost of $210 compared to Option 1 which costs $224.
Scenario 3: Budgeted $150 for visiting the park this summer
Option 1: Pay per visit
Maximum number of visits = Budgeted amount ÷ Cost per visit
Maximum number of visits = $150 ÷ $28 per visit
Maximum number of visits = 5.36 (rounded down to 5 visits)
Option 2: Purchase monthly membership
Total cost = Cost of membership + (Cost per visit × Maximum number of visits)
Total cost = $90 + ($15 × 5 visits)
Total cost = $90 + $75
Total cost = $165
In this scenario, Option 1 (pay per visit) is the best choice as it allows for a maximum of 5 visits within the budgeted amount, while Option 2 would exceed the budgeted amount.
Break-even point: Cost of both options would be approximately the same
Let's assume the break-even point is "x" visits.
Cost of Option 1: Total cost = $28 per visit × x visits
Cost of Option 2: Total cost = $90 + ($15 × x visits)
Setting these two equations equal to each other:
$28x = $90 + $15x
Rearranging the equation:
$28x - $15x = $90
$13x = $90
Solving for x:
x = $90 / $13
x ≈ 6.92
Therefore, the approximate break-even point would be around 7 visits.
To determine which option is best, we need to compare the total cost of each option for different scenarios. Let's calculate the costs for each option in each scenario using the given information.
1. Scenario 1: You plan to attend the park 3 times this summer.
Option 1: Pay per visit
Cost per visit: $28
Total cost: $28 * 3 = $84
Option 2: Purchase monthly membership and pay per visit
Monthly membership cost: $90
Cost per visit: $15
Total cost (monthly membership + 3 visits): $90 + ($15 * 3) = $135
In this scenario, the "pay per visit" option is cheaper, as the total cost is $84 compared to $135 with the monthly membership option.
2. Scenario 2: You expect to visit the park about 8 times this summer.
Option 1: Pay per visit
Cost per visit: $28
Total cost: $28 * 8 = $224
Option 2: Purchase monthly membership and pay per visit
Monthly membership cost: $90
Cost per visit: $15
Total cost (monthly membership + 8 visits): $90 + ($15 * 8) = $210
In this scenario, the "monthly membership and pay per visit" option is cheaper, as the total cost is $210 compared to $224 with the "pay per visit" option.
3. Scenario 3: You have budgeted $150 for visiting the park this summer.
Option 1: Pay per visit
Cost per visit: $28
Maximum number of visits within budget: $150 / $28 = 5.36 (approx.)
Since visits cannot be fractional, you can visit a maximum of 5 times.
Option 2: Purchase monthly membership and pay per visit
Monthly membership cost: $90
Cost per visit: $15
Maximum number of visits within budget: ($150 - $90) / $15 = 4 (approx.)
In this scenario, the "pay per visit" option is better as you can visit 5 times within your budget ($140) compared to 4 times with the monthly membership option.
4. Break-even point:
To find the break-even point where both options cost about the same, we can set the total cost of each option equal to each other and then solve for the number of visits.
Option 1: Pay per visit
Total cost = $28 * x (where x is the number of visits)
Option 2: Purchase monthly membership and pay per visit
Total cost = $90 + ($15 * x)
Setting the two equations equal to each other:
$28 * x = $90 + ($15 * x)
Solving for x:
$28x - $15x = $90
$13x = $90
x = $90 / $13 ≈ 6.92 (approx.)
Approximately, the break-even point is around 7 visits, where both options would cost about the same.
In summary, based on the given information:
- If you plan to attend the park 3 times, the "pay per visit" option is best.
- If you expect to visit the park about 8 times, the "monthly membership and pay per visit" option is best.
- If you have a budget of $150, the "pay per visit" option is best, allowing for up to 5 visits.
- The break-even point, where both options cost about the same, is around 7 visits.