A baseball team plays in a stadium that holds 52000 spectators. With the ticket price at $12 the average attendance has been 19000. When the price dropped to $9, the average attendance rose to 26000. Assume that attendance is linearly related to ticket price.
What ticket price would maximize revenue?

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- anonymous

P= price
x= attendance

- anonymous

x = kP +c for constants k and c

- anonymous

when P=12, x= 19000

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## More answers

- anonymous

when P= 9 , x= 26000

- anonymous

now you can form two eqns, solve for the constants c and k

- anonymous

19000 =12k +c
26000= 9k +c
subtract the first from the second
7000= -3k --> k = -7000/3

- anonymous

sub that into one of the other eqns
say the first , so c= 19000-12k = 47000

- anonymous

thats what ive done so far but nothing from there is making sense. I also have to describe what im doing in words so i prove i understand it.

- anonymous

so x= (-7000/3) P +47000

- anonymous

revenue = R = attendance times cost price

- anonymous

so R = P [ (-7000/3)P +47000 ]
differentiate with respect to P , we want to maximise R while varying P

- anonymous

R = (-7000/3) P^2 + 47000P
dR/dP = (-14000/3)P +47000 =0 for min/max

- anonymous

P = 3

- anonymous

now check, second derivative is d^2R/dP^2 = (-14000/3) which is less than zero , so we do get a maximum .
so ticket price = $3 maximises revenue

- anonymous

wit
its wrong

- anonymous

should be P = (-47000) / ( -14000/3 ) = $10.07 , that maximises revenue

- anonymous

$10.07 is the right answer. I still however have no idea how you got it. We haven't went over any problems by differentiating. I have no idea what that is.

- anonymous

well , this can be done by other methods , I am assuming you have done quadratic equations before , ie , you know how to find the vertex of a parabola?

- anonymous

and you know that the vertex ( which is a maximum or a minimum ) occurs on the axis of symmetry , and axis of symmetry of y= ax^2 +bx +c is x=-b / (2a)

- anonymous

yes that's what we're doing now

- anonymous

how did I know ;)

- anonymous

so if you go back to this eqn
R = (-7000/3) P^2 + 47000P
and find the axis of symmetry , which is P = -b/ (2a) , this will give you the x coordinate of the maximum point

- anonymous

which is 10.07, the answer

- anonymous

thank you!

- anonymous

Ok i know how you got this answer but why is there no c in the ax^2+bx+c equation? and how did you get 2 P's in the equation R= (-7000/3)P^2 +47000P ?

- anonymous

"why is there no c? " , there just isnt
"where did the two Ps come from" , look at how revenue was calculated, revenue = price x number of people

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