Chapter 4.1, Problem 4GP

To determine

To find the entry fee that should be to maximize the value of the pool.

Answer to Problem 4GP

  $\$65$

Explanation of Solution

Given:

Eddie has $120$ players and the entry fee is $\$40$ . Each time the fee is increased by $\$5$ , $10$ players are lost.

Calculation:

There are two Arithmetic progressions from the above information,

One is increasing AP as

  $\$40,45,50,\mathrm{...............}$

And other is decreasing AP

  $120,110,100,\mathrm{...............}$

The number of players will be zero when

  $\begin{array}{l}a+(n-1)d=0\\ 120+(n-1)(-10)=0\\ (n-1)=12\\ n=13\end{array}$

Now let us find out the entry fee as follows

  $\begin{array}{l}{a}_n=a+(n-1)d\\ a_{13}=5+(13-1)(5)\\ a_{13}=5+60\\ a_{13}=65\end{array}$

The entry fee should be $\$65$ .