Answer :
Let's analyze the costs using the given information according to the equations for each gym.
For Healthy Gym:
The total yearly cost ([tex]\(t\)[/tex]) is calculated by multiplying the fee per visit by the number of visits ([tex]\(n\)[/tex]):
[tex]\[ t_{\text{Healthy}} = 6n \][/tex]
For Fitness Gym:
The total yearly cost ([tex]\(t\)[/tex]) includes a one-time yearly charge plus a fee for each visit:
[tex]\[ t_{\text{Fitness}} = 50 + 5n \][/tex]
Now let's review each statement given in the context of these equations:
1. The total yearly cost is \[tex]$300 at each gym for 50 visits: - For Healthy Gym: \[ t_{\text{Healthy}} = 6 \times 50 = 300 \] - For Fitness Gym: \[ t_{\text{Fitness}} = 50 + 5 \times 50 = 50 + 250 = 300 \] This statement is true because both gyms will indeed cost \$[/tex]300 for 50 visits.
2. The total yearly cost will always be higher at Fitness Gym than at Healthy Gym:
- To determine the validity of this statement, let's consider different [tex]\(n\)[/tex]. For example, at 10 visits:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 10 = 60 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} > t_{\text{Healthy}} \][/tex]
However, if [tex]\(n\)[/tex] is sufficiently high, the cost of Fitness Gym might not always be higher due to the fixed component of Fitness Gym (the [tex]\( \$50 \)[/tex] yearly charge). For example, in higher numbers like [tex]\(n = 100\)[/tex]:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 100 = 600 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 100 = 50 + 500 = 550 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} < t_{\text{Healthy}} \][/tex]
This statement is false because the total yearly cost at Fitness Gym is not always higher than at Healthy Gym for all possible visit counts.
3. The total yearly cost is the same at both gyms when members make 10 visits to each one:
- For 10 visits:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 10 = 60 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} \ne t_{\text{Healthy}} \][/tex]
This statement is false because the costs are \[tex]$60 for Healthy Gym and \$[/tex]100 for Fitness Gym, which are not the same.
4. The total yearly cost at Healthy Gym is about \[tex]$50 lower than the total yearly cost at Fitness Gym: - With the provided values: - For Healthy Gym: \[ t_{\text{Healthy}} = 6 \times n \] - For Fitness Gym: \[ t_{\text{Fitness}} = 50 + 5 \times n \] When members make 10 visits: - Healthy Gym: \[ t_{\text{Healthy}} = 60 \] - Fitness Gym: \[ t_{\text{Fitness}} = 100 \] The difference: \[ t_{\text{Fitness}} - t_{\text{Healthy}} = 100 - 60 = 40 \] Comparing and considering other numbers of visits, the difference in yearly cost tends not to remain exactly \$[/tex]50 but might be around the associated fixed charge lower for Healthy Gymidn:t__. This statement is generally false, as it suggests an incorrect approximation.
Given the review of each statement based on the costs for the gyms, the first statement is the only one that is true:
The total yearly cost is \$300 at each gym for 50 visits.
For Healthy Gym:
The total yearly cost ([tex]\(t\)[/tex]) is calculated by multiplying the fee per visit by the number of visits ([tex]\(n\)[/tex]):
[tex]\[ t_{\text{Healthy}} = 6n \][/tex]
For Fitness Gym:
The total yearly cost ([tex]\(t\)[/tex]) includes a one-time yearly charge plus a fee for each visit:
[tex]\[ t_{\text{Fitness}} = 50 + 5n \][/tex]
Now let's review each statement given in the context of these equations:
1. The total yearly cost is \[tex]$300 at each gym for 50 visits: - For Healthy Gym: \[ t_{\text{Healthy}} = 6 \times 50 = 300 \] - For Fitness Gym: \[ t_{\text{Fitness}} = 50 + 5 \times 50 = 50 + 250 = 300 \] This statement is true because both gyms will indeed cost \$[/tex]300 for 50 visits.
2. The total yearly cost will always be higher at Fitness Gym than at Healthy Gym:
- To determine the validity of this statement, let's consider different [tex]\(n\)[/tex]. For example, at 10 visits:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 10 = 60 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} > t_{\text{Healthy}} \][/tex]
However, if [tex]\(n\)[/tex] is sufficiently high, the cost of Fitness Gym might not always be higher due to the fixed component of Fitness Gym (the [tex]\( \$50 \)[/tex] yearly charge). For example, in higher numbers like [tex]\(n = 100\)[/tex]:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 100 = 600 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 100 = 50 + 500 = 550 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} < t_{\text{Healthy}} \][/tex]
This statement is false because the total yearly cost at Fitness Gym is not always higher than at Healthy Gym for all possible visit counts.
3. The total yearly cost is the same at both gyms when members make 10 visits to each one:
- For 10 visits:
- Healthy Gym:
[tex]\[ t_{\text{Healthy}} = 6 \times 10 = 60 \][/tex]
- Fitness Gym:
[tex]\[ t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100 \][/tex]
Here:
[tex]\[ t_{\text{Fitness}} \ne t_{\text{Healthy}} \][/tex]
This statement is false because the costs are \[tex]$60 for Healthy Gym and \$[/tex]100 for Fitness Gym, which are not the same.
4. The total yearly cost at Healthy Gym is about \[tex]$50 lower than the total yearly cost at Fitness Gym: - With the provided values: - For Healthy Gym: \[ t_{\text{Healthy}} = 6 \times n \] - For Fitness Gym: \[ t_{\text{Fitness}} = 50 + 5 \times n \] When members make 10 visits: - Healthy Gym: \[ t_{\text{Healthy}} = 60 \] - Fitness Gym: \[ t_{\text{Fitness}} = 100 \] The difference: \[ t_{\text{Fitness}} - t_{\text{Healthy}} = 100 - 60 = 40 \] Comparing and considering other numbers of visits, the difference in yearly cost tends not to remain exactly \$[/tex]50 but might be around the associated fixed charge lower for Healthy Gymidn:t__. This statement is generally false, as it suggests an incorrect approximation.
Given the review of each statement based on the costs for the gyms, the first statement is the only one that is true:
The total yearly cost is \$300 at each gym for 50 visits.
