Answer:
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.
