Posted by Anonymous on Sunday, July 15, 2012 at 1:51pm.
A baseball team plays in a stadium that holds 53,000 spectators. With ticket prices at $10, the average attendance had been 49,000. When ticket prices were lowered to $8, the average attendance rose to 51,000.
(a) Find the demand function (price p as a function of attendance x), assuming it to be linear.
P(x)=_____
How should ticket prices be set to maximize revenue? (Round your answer to the nearest cent.)
$_________________

Calculus  Damon, Sunday, July 15, 2012 at 4:03pm
k is for thousands
(49 k, 10) and (51 k, 8)
slope = (810)/(51 k  49 k) = 1/1k
p = (1/k)x+ b
10 = (1/k)49 k + b
59 = b
so
p = (1/k)x + 59 or p = x/1000 + 59
r = x p
r = x^2/1000 + 59 x
x^2 59000 x = 1000 r
x^2  59000 x + 29,500^2 = 1000 r + 29,500^2
(x29,500)^2 = 1000(r + 870,250)
so max r at x = 29,500 people and revenue of 870,250
then price = 870,250/29,500 = $29.50
then p =

Calculus  Damon, Sunday, July 15, 2012 at 4:05pm
By the way there is no need to use calculus for these since you can complete the square to find the vertex of a parabola with algebra 2.

Calculus  Anonymous, Sunday, July 15, 2012 at 6:40pm
Oh okay, thank you!

Calculus  sign error (does not matter)  Damon, Sunday, July 15, 2012 at 6:49pm
r = x p
r = x^2/1000 + 59 x
x^2 59000 x = 1000 r
x^2  59000 x + 29,500^2 = 1000 r + 29,500^2
(x29,500)^2 = 1000(r  870,250)
so max r at x = 29,500 people and revenue of 870,250
then price = 870,250/29,500 = $29.50

Calculus  bob, Wednesday, October 17, 2012 at 11:54pm
i feel like you solved for when the stadium holds 59000 not 53000 like in his question hahaha WHICH IS PERFECT BECAUSE MY PROBLEM IS WITH 59000! yay :)
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