At the end of each round of a basketball tournament, half of the teams are eliminated from the competition. If the tournament starts with 64 teams, which equation shows how many rounds, r, must be completed to have 2 teams left in the tournament?
A.2(1/2)^r=64
B.2(2)^r/2=64
C.64(1/2)^r=2
D.64(2)^r=1/2
you are looking at a GP
with a =32, and r = 1/2
term(n) = a r^(n-1)
2 = 64(1/2)^(n-1)
1/32 = (1/2)^(n-1)
(1/2)^5 = (1/2)^(n-1)
n-1 = 5
n = 6
illustration:
64, 32, 16, 8, 4, 2 , count them you will get 6
A is obviously wrong, r = -5 makes no sense
B gives us r = 10 , silly
C gives us r = 5, but my listing shows that we have 4 teams after 5 rounds
D is absurd, the left side of the equation is 2048 when r = 5
none of your equations are correct, should be
64(1/2)^(r-1) = 2
hhm idk this answer, anyone out there know??
To find the equation that shows how many rounds, r, must be completed to have 2 teams left in the tournament, we need to think about the elimination process.
At the end of each round, half of the teams are eliminated. This means that after the first round, the number of teams remaining will be half of the original number of teams (64/2 = 32). After the second round, half of the remaining teams will be eliminated again, leaving 16 teams. This process continues until only 2 teams are left.
We can express this process as an equation. Let's break it down step by step.
After the first round, we have:
64/2 = 32 teams remaining
After the second round, we have:
32/2 = 16 teams remaining
After the third round, we have:
16/2 = 8 teams remaining
After the fourth round, we have:
8/2 = 4 teams remaining
After the fifth round, we have:
4/2 = 2 teams remaining
Based on this pattern, we can see that the number of teams remaining after each round is determined by dividing the previous number of teams remaining by 2. In other words, it is half of the previous number.
Now let's convert this pattern into an equation. Let r represent the number of rounds completed.
After the first round, the number of teams remaining is given by:
64/2 = 32
After the second round, the number of teams remaining is given by:
(64/2)/2 = 16
After the third round, the number of teams remaining is given by:
((64/2)/2)/2 = 8
After the fourth round, the number of teams remaining is given by:
(((64/2)/2)/2)/2 = 4
After the fifth round, the number of teams remaining is given by:
((((64/2)/2)/2)/2)/2 = 2
As we can see, we divide the original number of teams (64) by 2 repeatedly, r number of times. Therefore, the equation that represents this pattern is:
64/(2^r) = 2
Simplifying this equation gives us:
64 / (2^r) = 2
32 / (2^(r-1)) = 1
2^(r-1) = 32
2^(r-1) = 2^5
By comparing the exponents, we can see that r-1 = 5. Adding 1 to both sides gives us:
r = 6
Therefore, the correct equation that represents the number of rounds, r, required to have 2 teams left in the tournament is:
2^(r-1) = 32
Based on the provided options, none of them match this equation.