Chapter 6.4, Problem 48E

To determine

To write and solve an exponential equation to determine after which round there are $16$ terms left, if half the teams are eliminated after each round.

Answer to Problem 48E

  $T(n)=\frac{128}{2^n}$ is the required exponential equation that represents the number of teams left after $n\text{rounds}$ and after $3^{rd}$ round there are $16$ teams left.

Explanation of Solution

Given information:

Given there are a total of $128$ teams at the start of a citywide $3-\text{on}-3$ basketball tournament and half the teams are eliminated after each round.

Since we have $128$ teams at the start of basketball tournament and after each round half the teams are eliminated.

So after first round, the number of teams left $\frac{128}{2}$

Similar way after second round, the number of teams left $\frac{128}{2^2}$

In general after $n\text{rounds}$ , the number of teams left is $T(n)=\frac{128}{2^n}\mathrm{.........}(i)$

Thus, $T(n)=\frac{128}{2^n}$ is the required exponential equation that represents the number of teams left after $n\text{rounds}$ .

To determine after which round $(n)$ there are $16$ teams left, substitute $T(n)=16$ in equation $(i)$ , we get

  $\begin{array}{l}16=\frac{128}{2^n}\\ \Rightarrow 2^n=\frac{128}{16}\\ \Rightarrow 2^n=8\\ \Rightarrow 2^n=2^3\\ \therefore n=3\end{array}$

Therefore, after $3^{rd}$ round there are $16$ teams left.

Hence, $T(n)=\frac{128}{2^n}$ is the required exponential equation that represents the number of teams left after $n\text{rounds}$ and after $3^{rd}$ round there are $16$ teams left.