Question
Which could be represented by a linear function
A bank account that grows at a rate of 5% per year compounded annually
A population of bacteria that doubles every 4.5 hours
The cost of a cell phone service that charges a base amount plus 20 cents per minute
The concentration of medicine in a person’s body that decays by a factor of one third every hour
A bank account that grows at a rate of 5% per year compounded annually
A population of bacteria that doubles every 4.5 hours
The cost of a cell phone service that charges a base amount plus 20 cents per minute
The concentration of medicine in a person’s body that decays by a factor of one third every hour
Answers
GPT-4o mini
A linear function represents a relationship where the change in one variable results in a constant change in another variable. In other words, it describes a situation where the rate of change is constant.
Let's evaluate each option:
1. **A bank account that grows at a rate of 5% per year compounded annually**: This situation represents exponential growth, not linear growth, because the interest earned each year increases as the account balance increases. **Not linear**.
2. **A population of bacteria that doubles every 4.5 hours**: This situation is also exponential growth, as the population increases by a factor of 2 at regular intervals. **Not linear**.
3. **The cost of a cell phone service that charges a base amount plus 20 cents per minute**: This scenario can be represented as a linear function because the cost increases by a constant rate (20 cents) for each additional minute of service, plus a constant base amount. This is in the form \( C = b + 0.20m \), where \( C \) is the total cost, \( b \) is the base amount, and \( m \) is the number of minutes. **Linear**.
4. **The concentration of medicine in a person’s body that decays by a factor of one third every hour**: This situation describes exponential decay, as the concentration decreases by a fraction of its value rather than by a fixed amount. **Not linear**.
In summary, the only option that could be represented by a linear function is:
**The cost of a cell phone service that charges a base amount plus 20 cents per minute.**
Let's evaluate each option:
1. **A bank account that grows at a rate of 5% per year compounded annually**: This situation represents exponential growth, not linear growth, because the interest earned each year increases as the account balance increases. **Not linear**.
2. **A population of bacteria that doubles every 4.5 hours**: This situation is also exponential growth, as the population increases by a factor of 2 at regular intervals. **Not linear**.
3. **The cost of a cell phone service that charges a base amount plus 20 cents per minute**: This scenario can be represented as a linear function because the cost increases by a constant rate (20 cents) for each additional minute of service, plus a constant base amount. This is in the form \( C = b + 0.20m \), where \( C \) is the total cost, \( b \) is the base amount, and \( m \) is the number of minutes. **Linear**.
4. **The concentration of medicine in a person’s body that decays by a factor of one third every hour**: This situation describes exponential decay, as the concentration decreases by a fraction of its value rather than by a fixed amount. **Not linear**.
In summary, the only option that could be represented by a linear function is:
**The cost of a cell phone service that charges a base amount plus 20 cents per minute.**