Question
Two housing developments are building homes.
Development Description
The number of homes built increases by per year.
The number of homes built increases by per year.
Which statement is true?
Responses
The descriptions for both housing developments are best modeled by exponential functions.
The descriptions for both housing developments are best modeled by exponential functions.
The descriptions for both housing developments are best modeled by linear functions.
The descriptions for both housing developments are best modeled by linear functions.
The description for Development is best modeled by a linear function, while the description for Development is best modeled by an exponential function.
The description for Development 1 is best modeled by a linear function, while the description for Development 2 is best modeled by an exponential function.
The description for Development is best modeled by an exponential function, while the description for Development is best modeled by a linear function.
The description for Development 1 is best modeled by an exponential function, while the description for Development 2 is best modeled by a linear function.
Development Description
The number of homes built increases by per year.
The number of homes built increases by per year.
Which statement is true?
Responses
The descriptions for both housing developments are best modeled by exponential functions.
The descriptions for both housing developments are best modeled by exponential functions.
The descriptions for both housing developments are best modeled by linear functions.
The descriptions for both housing developments are best modeled by linear functions.
The description for Development is best modeled by a linear function, while the description for Development is best modeled by an exponential function.
The description for Development 1 is best modeled by a linear function, while the description for Development 2 is best modeled by an exponential function.
The description for Development is best modeled by an exponential function, while the description for Development is best modeled by a linear function.
The description for Development 1 is best modeled by an exponential function, while the description for Development 2 is best modeled by a linear function.
Answers
GPT-4o mini
To determine which statement is true regarding the housing developments, we need to consider what it means for the number of homes built to increase by a certain amount per year.
If the number of homes built increases by a fixed amount each year, this situation is best modeled by a linear function. For example, if Development 1 builds 100 homes in the first year and then builds the same number of homes in each subsequent year (e.g., 100 each year), that would result in a linear increase (100, 200, 300, etc.).
In contrast, if the description suggested that the number of homes built increases by a percentage of the current amount each year (like adding 10% more homes each subsequent year), that would be modeled by an exponential function.
From your wording, it is not explicitly stated how many homes are being built or how the number of homes is increasing for each development. However, based on standard interpretations in such scenarios:
- If both developments are described as increasing by a fixed number of homes each year, they are both linear.
- If one is increasing by a fixed number while the other is increasing by a percentage of the previous year's total, then we have a mix of linear and exponential.
Without more specific details about each development's numerical increase, we typically presume a fixed rate of increase leads to linear functions.
Based on this reasoning, the correct response is:
**The descriptions for both housing developments are best modeled by linear functions.**
If the number of homes built increases by a fixed amount each year, this situation is best modeled by a linear function. For example, if Development 1 builds 100 homes in the first year and then builds the same number of homes in each subsequent year (e.g., 100 each year), that would result in a linear increase (100, 200, 300, etc.).
In contrast, if the description suggested that the number of homes built increases by a percentage of the current amount each year (like adding 10% more homes each subsequent year), that would be modeled by an exponential function.
From your wording, it is not explicitly stated how many homes are being built or how the number of homes is increasing for each development. However, based on standard interpretations in such scenarios:
- If both developments are described as increasing by a fixed number of homes each year, they are both linear.
- If one is increasing by a fixed number while the other is increasing by a percentage of the previous year's total, then we have a mix of linear and exponential.
Without more specific details about each development's numerical increase, we typically presume a fixed rate of increase leads to linear functions.
Based on this reasoning, the correct response is:
**The descriptions for both housing developments are best modeled by linear functions.**