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Tatiana [17]
3 years ago
6

The average ticket price for a concert at the opera house was $50. The average attendance was 4000. When the ticket price was ra

ised to $52, attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize the revenue for the opera house?
Mathematics
1 answer:
kotegsom [21]3 years ago
6 0

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

p-50 = \dfrac{50-52}{4000-3800}(x - 4000)

p-50 = \dfrac{-2}{200}(x - 4000)

p-50 = \dfrac{-x}{100}+ 40

p = \dfrac{-x}{100}+ 90

R(x) = x . p

R(x) = x (\dfrac{-x}{100}+ 90)

R(x) = \dfrac{-x^2}{100}+ 90x)

\dfrac{d}{dx}(R(x)) = \dfrac{d}{dx}(\dfrac{-x^2}{100}+ 90x))

\dfrac{d}{dx}(R(x)) = (\dfrac{-2x}{100})+90)

at \dfrac{d}{dx}(R(x)) = 0

\dfrac{-2x}{100}= -90

x = 4500

\dfrac{d^2}{d^2x}(R(x)) = -ve

hence at x =4500 the revenue is maximum

for maximum revenue ticket price will be

p = \dfrac{-4500}{100}+ 90

p = $45

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