The number of teams remaining in a volleyball tournament can be determined by the function T parentheses n equals 32 parentheses one half exponent n where n is the number of completed round

Question

which statements are true select all that apply

The tournament started with 32 teams
two teams were eliminated in each round
half of the remaining team were eliminated each round.
There were 16 teams remaining after the first round
tournament lasted for five rounds.
Tournament lasted for six rounds.

Answer 1

We can analyze the function $T(n)=32\times {\left(\frac{1}{2}\right)}^n$ to determine the number of teams remaining after each round $n$.

1. The tournament started with 32 teams:
   True. When $n=0$ (before any rounds), $T(0)=32\times {\left(\frac{1}{2}\right)}^0=32$.

2. Two teams were eliminated in each round:
   False. In each round, half of the remaining teams are eliminated, not necessarily two. For example, after the first round, 16 teams remain, which implies 16 (not 2) were eliminated.

3. Half of the remaining teams were eliminated each round:
   True. The function shows that the number of teams remaining is half the previous round's total, i.e., $T(n)=\frac{1}{2}T(n-1)$.

4. There were 16 teams remaining after the first round:
   True. When $n=1$, $T(1)=32\times {\left(\frac{1}{2}\right)}^1=16$.

5. The tournament lasted for five rounds:
   False. After 5 rounds, $T(5)=32\times {\left(\frac{1}{2}\right)}^5=1$, which means only one team remains.

6. The tournament lasted for six rounds:
   True. After 6 rounds, $T(6)=32\times {\left(\frac{1}{2}\right)}^6=0.5$, indicating that the tournament can have logically concluded with one team remaining.

Based on this analysis, the true statements are:

• The tournament started with 32 teams.
• Half of the remaining teams were eliminated each round.
• There were 16 teams remaining after the first round.
• The tournament lasted for six rounds.