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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

Anya is traveling from out of town. This is the only time she will visit this trampoline park. Which option should she choose?
Jin plans on going to the trampoline park seven times this month. Which option should he choose? What does the point of intersection of the graphs represent?

1 answer

To evaluate whether to buy the membership or pay per visit, we can compare the costs associated with each option based on the number of visits \( x \).

Option A: The cost for paying per visit is represented by the equation:
\[ y = 9x \]
This means the total cost is $9 for each visit.

Option B: The cost for the monthly membership is represented by the equation:
\[ y = 30 + 4x \]
This means it costs $30 for the membership plus $4 for each visit.

To determine when it may be cheaper to purchase the membership, we can find the point of intersection of the two equations by setting them equal to each other:
\[ 9x = 30 + 4x \]

Now, we solve for \( x \):
\[ 9x - 4x = 30 \]
\[ 5x = 30 \]
\[ x = 6 \]

This means that the two options will cost the same when making 6 visits. Let's analyze the costs for fewer and more visits:

  1. If \( x < 6 \) (when making fewer than 6 visits):

    • Option A (pay per visit): \( y = 9x \)
    • Option B (monthly membership): \( y = 30 + 4x \)
    • For example:
      • \( x = 1 \):
        • Option A: \( 9(1) = 9 \)
        • Option B: \( 30 + 4(1) = 34 \) -> Option A is cheaper.
      • \( x = 5 \):
        • Option A: \( 9(5) = 45 \)
        • Option B: \( 30 + 4(5) = 50 \) -> Option A is cheaper.

  2. If \( x > 6 \) (when making more than 6 visits):

    • Option A (pay per visit): \( y = 9x \)
    • Option B (monthly membership): \( y = 30 + 4x \)
    • For example:
      • \( x = 7 \):
        • Option A: \( 9(7) = 63 \)
        • Option B: \( 30 + 4(7) = 58 \) -> Option B is cheaper.
      • \( x = 10 \):
        • Option A: \( 9(10) = 90 \)
        • Option B: \( 30 + 4(10) = 70 \) -> Option B is cheaper.

Choices for Anya and Jin:

  • Anya is traveling from out of town and will only visit once.

    • For \( x = 1 \):
      • Option A: \( y = 9(1) = 9 \)
      • Option B: \( y = 30 + 4(1) = 34 \)
    • Anya should choose Option A since it is cheaper for just one visit.

  • Jin plans to go seven times this month.

    • For \( x = 7 \):
      • Option A: \( y = 9(7) = 63 \)
      • Option B: \( y = 30 + 4(7) = 58 \)
    • Jin should choose Option B since it is cheaper for seven visits.

Point of Intersection:

The point of intersection of the graphs represents the number of visits (\( x \)) at which both options cost the same. In this case, the point of intersection is at \( (6, 54) \), meaning that at 6 visits, both options will cost $54. This point can help a customer decide which option is preferable based on their expected number of visits. If they plan to visit fewer than 6 times, option A is preferable; if they plan to visit more than 6 times, option B is preferable.