Question

A soccer stadium holds 62,000 spectators. With a ticket price of $20, the average attendance has been 24,000. When the price dropped to $16, the average attendance rose to 29,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue? Please show work

Answer 1

Two points are ( 24000 , 20) and ( 29000, 16), find the linear equation as

slope = ( 16 -20)/(29000 -24000)

= -1/1250

=-0.0008

Find the eqaution as

p - 20 = -0.0008 (x-24000 )

p = -0.0008x + 24000 *0.0008 +20

p = -0.0008x +39.2

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Find the revenue as

R=x*p

= -0.0008x² +39.2x

Find the derivative as

R' = -0.0008(2x) +39.2

Set R' =0 and then solve for x as

-0.0008(2x) +39.2 = 0

x = 39.2/(2*0.0008)

x = 24500

Find price as

p = -0.0008( 24500) +39.2

=$19.6

Hence, ticket price would maximize revenue is $19.6