Question
The number of teams remaining in a tournament decreases as each round of the tournament is completed. The function models the total number of teams remaining after rounds. Which statement about the function is true?
Responses
The number 256 represents the decay rate of the function.
The number 256 represents the decay rate of the function.
The number 256 represents the growth rate of the function.
The number 256 represents the growth rate of the function.
The expression represents the decay rate of the function.
The expression 1 minus b represents the decay rate of the function.
The expression represents the growth rate of the function.
Responses
The number 256 represents the decay rate of the function.
The number 256 represents the decay rate of the function.
The number 256 represents the growth rate of the function.
The number 256 represents the growth rate of the function.
The expression represents the decay rate of the function.
The expression 1 minus b represents the decay rate of the function.
The expression represents the growth rate of the function.
Answers
GPT-4o mini
To analyze the statements correctly, it's important to understand the nature of decay and growth in functions, particularly in relation to tournaments.
If a function models the number of teams remaining after each round and it decreases, then we're dealing with a decay function. Generally, in such contexts:
1. **Decay Rate**: The decay rate can often be expressed as \( b \), where \( 0 < b < 1 \), representing the fraction remaining after each round. The expression \( 1 - b \) would then be the proportion of teams that are eliminated after each round.
2. **Growth Rate**: In the context of decay functions, a growth rate refers to an increase, which would not apply here since the number of teams is decreasing.
Given the options:
- "The number 256 represents the decay rate of the function." — This is unlikely unless the function specifically indicates that.
- "The number 256 represents the growth rate of the function." — This is false; a growth rate would not apply here.
- "The expression \( 1 - b \) represents the decay rate of the function." — Correct, as it captures the proportion of teams eliminated.
- "The expression \( b \) represents the growth rate of the function." — False in a decay context.
Based on this analysis, the true statement is:
**The expression \( 1 - b \) represents the decay rate of the function.**
If a function models the number of teams remaining after each round and it decreases, then we're dealing with a decay function. Generally, in such contexts:
1. **Decay Rate**: The decay rate can often be expressed as \( b \), where \( 0 < b < 1 \), representing the fraction remaining after each round. The expression \( 1 - b \) would then be the proportion of teams that are eliminated after each round.
2. **Growth Rate**: In the context of decay functions, a growth rate refers to an increase, which would not apply here since the number of teams is decreasing.
Given the options:
- "The number 256 represents the decay rate of the function." — This is unlikely unless the function specifically indicates that.
- "The number 256 represents the growth rate of the function." — This is false; a growth rate would not apply here.
- "The expression \( 1 - b \) represents the decay rate of the function." — Correct, as it captures the proportion of teams eliminated.
- "The expression \( b \) represents the growth rate of the function." — False in a decay context.
Based on this analysis, the true statement is:
**The expression \( 1 - b \) represents the decay rate of the function.**