Answer
given,
opera house ticket = $50
attendance = 4000 persons
now,
opera house ticket = $52
attendance = 3800 person
assuming these are the points on the demand curve
(x, p) = (4000,50) and (x,p) = (3800,52)
using point slope formula
[tex]p-50 = \dfrac{50-52}{4000-3800}(x - 4000)[/tex]
[tex]p-50 = \dfrac{-2}{200}(x - 4000)[/tex]
[tex]p-50 = \dfrac{-x}{100}+ 40[/tex]
[tex]p = \dfrac{-x}{100}+ 90[/tex]
R(x) = x . p
[tex]R(x) = x (\dfrac{-x}{100}+ 90)[/tex]
[tex]R(x) = \dfrac{-x^2}{100}+ 90x)[/tex]
[tex]\dfrac{d}{dx}(R(x)) = \dfrac{d}{dx}(\dfrac{-x^2}{100}+ 90x))[/tex]
[tex]\dfrac{d}{dx}(R(x)) = (\dfrac{-2x}{100})+90)[/tex]
at [tex]\dfrac{d}{dx}(R(x)) = 0[/tex]
[tex]\dfrac{-2x}{100}= -90[/tex]
x = 4500
[tex]\dfrac{d^2}{d^2x}(R(x)) = -ve[/tex]
hence at x =4500 the revenue is maximum
for maximum revenue ticket price will be
[tex]p = \dfrac{-4500}{100}+ 90[/tex]
p = $45
