The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the price of the ticket is $50$ dollars. A market survey indicates that $10$ additional seats will remain empty for each $5$ dollar increase of the ticket price. What is the ticket price which maximizes the manager's revenue? How many seats will be occupied at that price?
To solve this problem, I think I have to maximize $f(t)=(50+5t)(1000-10t)$ where $t$ ranges from $0$ to $100$. Therefore the answer is $275$ dollars and $550$ seats. Am I correct? Could anyone please check for me?