Question
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: 50+5n=t Which statement about the total yearly cost of the memberships is true? (1 point) Responses The total yearly cost is the same at both gyms when members make 10 visits to each one. The total yearly cost is the same at both gyms when members make 10 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 at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym. The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym. The total yearly cost is $300 at each gym for 50 visits. The total yearly cost is $300 at each gym for 50 visits.
Answers
GPT-4o mini
To find the total yearly cost at each gym based on the number of visits \( n \), we can substitute the values into the equations provided.
For Healthy Gym:
\[ t = 6n \]
For Fitness Gym:
\[ t = 50 + 5n \]
Now let's evaluate the options provided.
1. **When \( n = 10 \):**
- Healthy Gym:
\[ t = 6 \times 10 = 60 \]
- Fitness Gym:
\[ t = 50 + 5 \times 10 = 50 + 50 = 100 \]
- The total yearly costs are not equal.
2. **Total cost comparison:**
- As \( n \) increases, the cost at Healthy Gym increases linearly at a rate of 6 per visit, while the cost at Fitness Gym starts at $50 and increases at a rate of 5 per visit.
- Initially, when \( n = 0 \), the cost at Fitness Gym is already higher due to the $50 one-time charge. However, at higher values of \( n \), Healthy Gym will eventually become more expensive than Fitness Gym.
3. **When solving for \( n \) where costs equal:**
Set the two equations equal to find out how many visits lead to the same total cost:
\[ 6n = 50 + 5n \]
\[ 6n - 5n = 50 \]
\[ n = 50 \]
When \( n = 50 \):
- Healthy Gym:
\[ t = 6 \times 50 = 300 \]
- Fitness Gym:
\[ t = 50 + 5 \times 50 = 50 + 250 = 300 \]
Thus, the total yearly cost is the same at both gyms when members make 50 visits, not 10.
Considering these evaluations, the only statement that is true is:
**The total yearly cost is $300 at each gym for 50 visits.**
For Healthy Gym:
\[ t = 6n \]
For Fitness Gym:
\[ t = 50 + 5n \]
Now let's evaluate the options provided.
1. **When \( n = 10 \):**
- Healthy Gym:
\[ t = 6 \times 10 = 60 \]
- Fitness Gym:
\[ t = 50 + 5 \times 10 = 50 + 50 = 100 \]
- The total yearly costs are not equal.
2. **Total cost comparison:**
- As \( n \) increases, the cost at Healthy Gym increases linearly at a rate of 6 per visit, while the cost at Fitness Gym starts at $50 and increases at a rate of 5 per visit.
- Initially, when \( n = 0 \), the cost at Fitness Gym is already higher due to the $50 one-time charge. However, at higher values of \( n \), Healthy Gym will eventually become more expensive than Fitness Gym.
3. **When solving for \( n \) where costs equal:**
Set the two equations equal to find out how many visits lead to the same total cost:
\[ 6n = 50 + 5n \]
\[ 6n - 5n = 50 \]
\[ n = 50 \]
When \( n = 50 \):
- Healthy Gym:
\[ t = 6 \times 50 = 300 \]
- Fitness Gym:
\[ t = 50 + 5 \times 50 = 50 + 250 = 300 \]
Thus, the total yearly cost is the same at both gyms when members make 50 visits, not 10.
Considering these evaluations, the only statement that is true is:
**The total yearly cost is $300 at each gym for 50 visits.**