Question

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?

Answer 1

P= price

x= attendance

Answer 2

x = kP +c for constants k and c

Answer 3

when P=12, x= 19000

Answer 4

when P= 9 , x= 26000

Answer 5

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

Answer 6

19000 =12k +c

26000= 9k +c

subtract the first from the second

7000= -3k --> k = -7000/3

Answer 7

sub that into one of the other eqns

say the first , so c= 19000-12k = 47000

Answer 8

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.

Answer 9

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

Answer 10

revenue = R = attendance times cost price

Answer 11

so R = P [ (-7000/3)P +47000 ]

differentiate with respect to P , we want to maximise R while varying P

Answer 12

R = (-7000/3) P^2 + 47000P

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

Answer 13

P = 3

Answer 14

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

Answer 15

wit
its wrong

Answer 16

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

Answer 17

$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.

Answer 18

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?

Answer 19

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)

Answer 20

yes that's what we're doing now

Answer 21

how did I know ;)

Answer 22

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

Answer 23

which is 10.07, the answer

Answer 24

thank you!

Answer 25

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 ?

Answer 26

"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