Question

a chess tournament starts with 1024 players. after each round, half of the players are eliminated.
a. write an equation that represents the number of players, p, that remain after r rounds, and explain how you determined your answer.
b. how many players will there be remaining after 6 rounds?

Explanation:

Step1: Identify the pattern

The initial number of players is 1024. After each round, the number of players is multiplied by $\frac{1}{2}$. This is an exponential - decay model. The general form of an exponential function is $P = a\cdot b^{r}$, where $a$ is the initial amount, $b$ is the common ratio, and $r$ is the number of time - periods. Here, $a = 1024$ and $b=\frac{1}{2}$. So the equation is $P = 1024\cdot(\frac{1}{2})^{r}$.

Step2: Substitute $r = 6$ into the equation

We want to find the number of players remaining after 6 rounds. Substitute $r = 6$ into the equation $P = 1024\cdot(\frac{1}{2})^{r}$. First, we know that $1024 = 2^{10}$ and $(\frac{1}{2})^{r}=2^{-r}$. So $P = 2^{10}\cdot2^{-6}$.

Step3: Use the exponent rule $a^{m}\cdot a^{n}=a^{m + n}$

According to the rule $a^{m}\cdot a^{n}=a^{m + n}$, we have $P=2^{10+( - 6)}=2^{4}$.

Step4: Calculate the value of $2^{4}$

$2^{4}=16$.

Answer:

A. The equation is $P = 1024\cdot(\frac{1}{2})^{r}$. We determined it based on the exponential - decay model where the initial number of players is 1024 and the common ratio of decrease after each round is $\frac{1}{2}$.
B. 16