The entry fee is $11, and the entrants make the recipes at home and vote for their favorite, which cannot be their own recipe. The top 2 finishers win money. First place wins $58, and second place wins $48. You give yourself an X chance of finishing in the top 2, and if you finish top 2, then you have an equal chance of finishing first or second. Required:
What is x such that entering this tournament is a fair gamble for you as an expected value maximizer?

The value is  

Step-by-step explanation:

From the question we are told that

    The entry fee is  

    The  outcome for  first place is  

    The outcome for second place is

   The outcome for losing is  

The chance of finishing in the top two  is  X

Given that there are equal chance  of finishing first or second, then the chance of finishing first is  

and  the chance of finishing second is  

Then the chance of losing(i.e not finishing in the first two) is  

given that the game is a fair gamble for the player as an expected maximizer then it mean is that the expected value of entering the tournament  E(X) =  0

So

=>  

=>  

=>  

=>  

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.

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step-by-step explanation: