Question

A large online video game tournament begins with 65,53665,536 teams. The number of teams, t,t, remaining after each round, r,r, can be expressed as t=65,536(12)r.t=65,536(12)r. Eight teams will advance to the quarterfinals. The number of rounds necessary for there to be 88 teams left can be modeled as r=log(1k)log(12).r=log⁡(1k)log⁡(12). What is the value of k?k?

Answer 1

k=8192

Explanation:

The number of teams,t remaining after each round, r, can be expressed as:

`t=65,536((1)/(2))^r`

• 8 Teams will advance to the quarterfinals.

First, we determine the round,r at which there will be 8 teams left.

`t=65,536((1)/(2))^r\\8=65536*0.5^r\\0.5^r=8 / 65536\\2^(-1r)=2^(-13)\\-r=-13\\r=13`

Using this value of r

`If \: r=(Log(1)/(k))/(Log(1)/(2)) \\Since\: r=13\\13=(Log(1)/(k))/(Log(1)/(2))\\$Cross Multiply$\\Log(1)/(k)=13 X Log 0.5\\ $Using a Log b=Log $b^(a)\\Log(1)/(k)= Log 0.5^(13)\\(1)/(k)=0.5^(13)\\(1)/(k)=(1)/(8192)\\k=8192`