The average ticket price for a concert at the opera house was $50. The average attendance was 2500. When the ticket price was raised to $54, attendance declined to an average of 2100 persons per performance. What should the ticket price be to maximize revenue for the opera house?

The price per ticket should be $37.5

Explanation:

First we need to determine the change in demand (attendance) as a result of every $1 increase in the price of ticket.

The ticket price increased by $4 (from 50 to 54) and the demand fell by 400 (from 2500 to 2100). The change per dollar is,  400 / 4 = 100.

So, for every $1 increase in price, demand falls by 100.

The revenue is calculated by multiplying price by quantity demanded. Revenue equation will be,

Let x be the change in price from $50.

Revenue = (50 + x)  * (2500 - 100x)

Revenue = 125000 - 5000x + 2500x - 100x²

Revenue = 125000 - 2500x - 100x²

To calculate the price that maximizes the revenue, we need to take the derivative of this equation.

d/dx = 0 - 1 * 2500x° - 2 * 100x

0 = -2500  -  200x

2500 = -200x

2500 / -200 = x

-12.5 = x

Price should be 50 - 12.5 = 37.5

At price $37.5 the revenue of the Opera House is maximized.

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