Question
A trampoline park that you frequently go to is $9 per visit. You have the option to purchase a monthly membership for $30 and then pay $4 for each visit. Explain whether you will buy the membership, and why. Option A: y = $9x Option B: y = $30 + $4x
Answers
GPT-4o mini
To determine whether purchasing the monthly membership is a better option than paying per visit, we can set up the equations based on your usage.
**Option A: Pay per visit.**
The cost function is:
\[
y = 9x
\]
where \( y \) is the total cost and \( x \) is the number of visits.
**Option B: Monthly membership.**
The cost function is:
\[
y = 30 + 4x
\]
where \( y \) is the total cost and \( x \) is the number of visits.
Now, we can compare the two cost functions to find the break-even point, which is when both costs are equal:
\[
9x = 30 + 4x
\]
To solve for \( x \):
1. Subtract \( 4x \) from both sides:
\[
9x - 4x = 30
\]
\[
5x = 30
\]
2. Divide by 5:
\[
x = 6
\]
This means that if you plan to visit the trampoline park more than 6 times in a month, the membership option will be cheaper.
- **If you visit 6 times:**
- Option A: \( 9 \times 6 = 54 \)
- Option B: \( 30 + 4 \times 6 = 30 + 24 = 54 \) (break-even)
- **If you visit 7 times:**
- Option A: \( 9 \times 7 = 63 \)
- Option B: \( 30 + 4 \times 7 = 30 + 28 = 58 \) ( membership is cheaper)
- **If you visit 5 times:**
- Option A: \( 9 \times 5 = 45 \)
- Option B: \( 30 + 4 \times 5 = 30 + 20 = 50 \) (pay per visit is cheaper)
Given this analysis, it would make sense to buy the membership if you plan to visit more than 6 times in a month. If you anticipate using the trampoline park less than or equal to 6 times, then it would be more cost-effective to pay the regular fee of $9 per visit.
**Option A: Pay per visit.**
The cost function is:
\[
y = 9x
\]
where \( y \) is the total cost and \( x \) is the number of visits.
**Option B: Monthly membership.**
The cost function is:
\[
y = 30 + 4x
\]
where \( y \) is the total cost and \( x \) is the number of visits.
Now, we can compare the two cost functions to find the break-even point, which is when both costs are equal:
\[
9x = 30 + 4x
\]
To solve for \( x \):
1. Subtract \( 4x \) from both sides:
\[
9x - 4x = 30
\]
\[
5x = 30
\]
2. Divide by 5:
\[
x = 6
\]
This means that if you plan to visit the trampoline park more than 6 times in a month, the membership option will be cheaper.
- **If you visit 6 times:**
- Option A: \( 9 \times 6 = 54 \)
- Option B: \( 30 + 4 \times 6 = 30 + 24 = 54 \) (break-even)
- **If you visit 7 times:**
- Option A: \( 9 \times 7 = 63 \)
- Option B: \( 30 + 4 \times 7 = 30 + 28 = 58 \) ( membership is cheaper)
- **If you visit 5 times:**
- Option A: \( 9 \times 5 = 45 \)
- Option B: \( 30 + 4 \times 5 = 30 + 20 = 50 \) (pay per visit is cheaper)
Given this analysis, it would make sense to buy the membership if you plan to visit more than 6 times in a month. If you anticipate using the trampoline park less than or equal to 6 times, then it would be more cost-effective to pay the regular fee of $9 per visit.