I really don't understand how to do this, please help

Question

A baseball team plays in a stadium that holds 56000 spectators. With the ticket price at $9 the average attendance has been 25000. When the price dropped to $8, the average attendance rose to 28000.

a) Find the demand function , where  is the number of the spectators. (Assume that  is linear.)

 b) How should ticket prices be set to maximize revenue?The revenue is maximized by charging $ per ticket.

Answer 1

The demand Q is a function of price p...

(9,25000)

(8,28000)

so the slope of the demand line is -3000; that is the demand decreases by 3000

when the price increases by $1

B = y - mx = 25000 - (-3000)(9) = 25000 + 27000 = 52000

the linear demand function is Q(p) = -3000p + 52000

the Revenue function is then R(p) = p * Q(p) = p * (52000 - 3000p)

= -3000p^2 + 52000p

A=-3000 B = 52000

max revenue occurs at -B/(2a) = -52000/(-6000) = 52/6 = 26/3

$8.33