The greatest revenue?

Question

A theatre seats 2000 people and charges $10 for a ticket. At this price all the tickets can be sold. A survey indicates that if the ticket price is increased, the number sold will decrease by 100 for every dollar of increase. What ticket price would result in the greatest revenue?

Answer 1

To solve this problem, you need to use some simple calculus. Let $x$ be the price of the ticket. When $x=10$, we know that $2000$ seats are sold and that $100$ fewer seats are sold for every dollar increase in price. This nets us the equation:

seats sold $= 2000-100(x-10)$

If we multiply this by the price of the ticket, we can find the total revenue $R$:

$R=x(2000-100(x-10))$

$=2000x-100x(x-10)$

$=2000x-100x^{2}+1000x$

$=3000x-100x^2$

Since we're trying to find a maximum for this expression, we can run this function for revenue through any monotonically increasing analytic function and their maximum will occur at the same value of $x$. Therefore, we can divide the equation by $100$ to find:

$30x-x^{2}$

Taking the derivative with respect to $x$, we have:

$30-2x$

... and setting this to zero, we have:

$0=30-2x$

$2x=30$

$x=15$

Lastly, we can verify that this is a maximum and not a minimum because the second derivative of our revenue function is negative.

Therefore, you'll want to price your tickets at $15.

Answer 2

Hint: If we have $x$ dollars of increase, we get: $$(2000-100x)(10+x)$$ We are attempting to maximize this value.

Answer 3

Hint. Let the price be increased by $x$ dollars. Then the number of people would be $2000-100x$, and the price per ticket would be $10+x$. What would be the revenue as a function of $x$?