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Healthy Gym charges members a fee for each visit, while Fitness Gym charges members a one-time yearly charge plus a fee for each visit. The equations below represent the amounts members pay at each gym, where t represents the total yearly cost and n represents the number of visits.Healthy Gym: 6n=t Fitness Gym: 80+4n=t Which statement about the total yearly cost of the memberships is true? NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature twice per question.(1 point)ResponsesThe total yearly cost is the same at both gyms when members make 25 visits to each one.The total yearly cost is the same at both gyms when members make 25 visits to each one.The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.The total yearly cost is $240 at each gym for 40 visits.The total yearly cost is $240 at each gym for 40 visits.The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.Skip to navigation

1 answer

To determine the accurate statements about the total yearly cost of the memberships at Healthy Gym and Fitness Gym, we can analyze their equations:

  1. Healthy Gym: \[ t = 6n \]

  2. Fitness Gym: \[ t = 80 + 4n \]

Now we can evaluate the cost at different visit numbers.

When \( n = 25 \):

  • Healthy Gym: \[ t = 6(25) = 150 \]

  • Fitness Gym: \[ t = 80 + 4(25) = 80 + 100 = 180 \]

Conclusion: Total yearly cost is not the same at both gyms when members make 25 visits.

When \( n = 40 \):

  • Healthy Gym: \[ t = 6(40) = 240 \]

  • Fitness Gym: \[ t = 80 + 4(40) = 80 + 160 = 240 \]

Conclusion: The total yearly cost is the same at both gyms for 40 visits.

General Case Analysis:

Next, we can derive the conditions under which one gym would be costlier than the other.

Setting the total costs from both gyms equal allows us to find the crossover point: \[ 6n = 80 + 4n \]

Rearranging gives: \[ 6n - 4n = 80 \] \[ 2n = 80 \] \[ n = 40 \]

Beyond 40 Visits:

  • For \( n > 40 \): Healthy Gym will be cheaper since it costs $6 per visit compared to Fitness Gym's $4 per visit plus a fixed charge of $80.

Below 40 Visits:

  • For \( n < 40 \): Fitness Gym will be cheaper as the one-time fee becomes less significant with fewer visits.

Summary of Responses:

  1. The total yearly cost is the same at both gyms when members make 25 visits to each one. - False
  2. The total yearly cost will always be higher at Fitness Gym than at Healthy Gym. - False
  3. The total yearly cost is $240 at each gym for 40 visits. - True
  4. The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym. - False

Correct Statement:

The correct statement is: The total yearly cost is $240 at each gym for 40 visits.