find a demand function d(q) , where q is the quantity/number of the spectators. (assume d(q) is linear)

Question

a baseball team plays in a stadium that holds 72000 spectators. with the ticket price at $8 the average attendance has been 29000. when the price dropped to $4, the average attendance rose to 36000. find a demand function d(q) , where q is the quantity/number of the spectators. (assume d(q) is linear)

Answer 1

(29, 8) and (36, 4) are two points on a linear demand curve

slope= (8-4)/(29-36) = -4/7 = (y-4)/(x-36) where x=q measured in thousands, y=d(q)= price measured in dollars

y-4 = (-4/7)(x-36) in point slope form

y = d(q) = -4q/7 + 4(36)/7 +4

d(q) = -4q/7 + 40 in slope intercept form, the point (0,40) is the y intercept

graphically it's a line from (0,40) to (70, 0)

they can never fill the stadium to capacity of 72, without a negative price, paying customers to come

at a price of $40, zero ticket sales

Revenue = dq = -4q^2/7 +40q

Max Revenue is when (dq)' = -8q/7 +40 = 0

q = 40(7/8) = $35 = half of 70 ,

and at $35, ticket sales = half of 40 = 20,000

max Revenue = dq= 20(35) = 700 = $700,000 given units in thousands of dollars

the Revenue maximizing point is the midpoint of the linear demand curve, midway between the two intercepts