Question
There were 64 teams remaining in a tournament after the first round, which is 1/2
of the original number of teams. The remaining number of teams after each round can be represented by an exponential equation where y is the number of teams remaining after x rounds.
Which statement is TRUE about the model?
the equation modeling the scenario is y = 128(1/2)^x
. The points (0, 128) and (1, 64) are both valid data points for the model in this scenario.
The equation modeling the scenario is
. The points (1, 64) and
are both valid data points for the model in this scenario.
all the above
of the original number of teams. The remaining number of teams after each round can be represented by an exponential equation where y is the number of teams remaining after x rounds.
Which statement is TRUE about the model?
the equation modeling the scenario is y = 128(1/2)^x
. The points (0, 128) and (1, 64) are both valid data points for the model in this scenario.
The equation modeling the scenario is
. The points (1, 64) and
are both valid data points for the model in this scenario.
all the above
Answers
GPT-4o mini
Let's analyze the situation to determine which statements are true.
1. **Understanding the Tournament Structure:**
- After the first round, there are 64 teams remaining, which is half of the original number of teams. This means the original number of teams was 128 (since \( 64 \times 2 = 128 \)).
2. **Modeling the Number of Teams Remaining:**
- Every round, half of the teams are eliminated. Therefore, the number of teams remaining after each round forms an exponential decay pattern.
- The general form of the exponential decay can be expressed as:
\[
y = a(1/2)^x
\]
- Here, \( a \) is the initial number of teams, and \( x \) is the number of rounds.
3. **Setting Up the Equation:**
- From the information given:
- Initial number of teams \( a = 128 \)
- After 1 round (x=1), number of teams remaining \( y = 64 \).
Therefore, the equation modeling the number of teams remaining is:
\[
y = 128(1/2)^x
\]
4. **Verification of Points:**
- When \( x = 0 \):
\[
y = 128(1/2)^0 = 128(1) = 128
\]
- This means the point (0, 128) is valid.
- When \( x = 1 \):
\[
y = 128(1/2)^1 = 128(1/2) = 64
\]
- This gives the point (1, 64), which is also valid.
Now, checking the statements:
- **Statement 1:** "The equation modeling the scenario is \( y = 128(1/2)^x \). The points (0, 128) and (1, 64) are both valid data points for the model in this scenario."
- This statement is TRUE.
- **Statement 2:** "The equation modeling the scenario is [incomplete]. The points (1, 64) and [missing point] are both valid data points for the model in this scenario."
- This statement is incomplete but could be interpreted as focusing only on the second part. However, based on the provided details, it could imply other points which might also be valid (for instance, the next round would yield 32 at \( x=2 \)).
- **Statement 3:** "All the above."
- Since the first statement is true, and the second statement is valid contextually (but incomplete), this statement is also considered true in the context of the provided information.
In conclusion, the answer is that the first statement is clearly true, and we can conclude the implication of the second statement as valid in terms of the points provided. Thus:
**The correct option is: "all the above"** is considered true based on the analysis, but with the understanding that not all statements are equally complete.
1. **Understanding the Tournament Structure:**
- After the first round, there are 64 teams remaining, which is half of the original number of teams. This means the original number of teams was 128 (since \( 64 \times 2 = 128 \)).
2. **Modeling the Number of Teams Remaining:**
- Every round, half of the teams are eliminated. Therefore, the number of teams remaining after each round forms an exponential decay pattern.
- The general form of the exponential decay can be expressed as:
\[
y = a(1/2)^x
\]
- Here, \( a \) is the initial number of teams, and \( x \) is the number of rounds.
3. **Setting Up the Equation:**
- From the information given:
- Initial number of teams \( a = 128 \)
- After 1 round (x=1), number of teams remaining \( y = 64 \).
Therefore, the equation modeling the number of teams remaining is:
\[
y = 128(1/2)^x
\]
4. **Verification of Points:**
- When \( x = 0 \):
\[
y = 128(1/2)^0 = 128(1) = 128
\]
- This means the point (0, 128) is valid.
- When \( x = 1 \):
\[
y = 128(1/2)^1 = 128(1/2) = 64
\]
- This gives the point (1, 64), which is also valid.
Now, checking the statements:
- **Statement 1:** "The equation modeling the scenario is \( y = 128(1/2)^x \). The points (0, 128) and (1, 64) are both valid data points for the model in this scenario."
- This statement is TRUE.
- **Statement 2:** "The equation modeling the scenario is [incomplete]. The points (1, 64) and [missing point] are both valid data points for the model in this scenario."
- This statement is incomplete but could be interpreted as focusing only on the second part. However, based on the provided details, it could imply other points which might also be valid (for instance, the next round would yield 32 at \( x=2 \)).
- **Statement 3:** "All the above."
- Since the first statement is true, and the second statement is valid contextually (but incomplete), this statement is also considered true in the context of the provided information.
In conclusion, the answer is that the first statement is clearly true, and we can conclude the implication of the second statement as valid in terms of the points provided. Thus:
**The correct option is: "all the above"** is considered true based on the analysis, but with the understanding that not all statements are equally complete.