Question
After Cheer Coach Moore completed grade checks and verified other eligibility requirements, there 256 were individuals remaining to try out for the 8 positions on the Sunnyville High School Varsity Cheer Squad. The individuals will go through three weeks of training, then will be face several rounds of competition. After each round, half of the individuals will be eliminated from the try-outs.
8=__(___)^x
How many rounds are needed to go from the individuals trying out to the
individuals who will be on the Sunnyville High School Varsity Cheer Squad?
8=__(___)^x
How many rounds are needed to go from the individuals trying out to the
individuals who will be on the Sunnyville High School Varsity Cheer Squad?
Answers
GPT-4o mini
To determine how many rounds of elimination are needed to go from 256 individuals to 8 individuals, we can express this as an equation based on the elimination process.
Given that half of the individuals are eliminated in each round, we can represent the number of individuals remaining after \( x \) rounds with the equation:
\[
\text{Remaining individuals} = \frac{\text{Initial individuals}}{2^x}
\]
In this scenario, the initial number of individuals is 256, and we want to find \( x \) such that:
\[
\frac{256}{2^x} = 8
\]
To solve for \( x \), first multiply both sides by \( 2^x \):
\[
256 = 8 \cdot 2^x
\]
Next, we can rewrite 8 as \( 2^3 \):
\[
256 = 2^3 \cdot 2^x
\]
\[
256 = 2^{3+x}
\]
Now we need to express 256 in terms of a power of 2. We know that:
\[
256 = 2^8
\]
Now we have:
\[
2^8 = 2^{3+x}
\]
By equating the exponents, we set up the equation:
\[
8 = 3 + x
\]
Now, subtract 3 from both sides:
\[
x = 8 - 3
\]
\[
x = 5
\]
Thus, the number of rounds needed to go from 256 individuals to 8 individuals is **5 rounds**.
Given that half of the individuals are eliminated in each round, we can represent the number of individuals remaining after \( x \) rounds with the equation:
\[
\text{Remaining individuals} = \frac{\text{Initial individuals}}{2^x}
\]
In this scenario, the initial number of individuals is 256, and we want to find \( x \) such that:
\[
\frac{256}{2^x} = 8
\]
To solve for \( x \), first multiply both sides by \( 2^x \):
\[
256 = 8 \cdot 2^x
\]
Next, we can rewrite 8 as \( 2^3 \):
\[
256 = 2^3 \cdot 2^x
\]
\[
256 = 2^{3+x}
\]
Now we need to express 256 in terms of a power of 2. We know that:
\[
256 = 2^8
\]
Now we have:
\[
2^8 = 2^{3+x}
\]
By equating the exponents, we set up the equation:
\[
8 = 3 + x
\]
Now, subtract 3 from both sides:
\[
x = 8 - 3
\]
\[
x = 5
\]
Thus, the number of rounds needed to go from 256 individuals to 8 individuals is **5 rounds**.