maximizing income and quadratic function

Question

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?

Answer 1

$ f(t)\,=(50\,+5t)\,(1000-\,10t) $ Now f'(t)=5(1000 - 10t) - 10 (50 + 5t), Where f'(t)=0 at t= 45 and f''(t)=-50-50 <0. Hence has maximum at t =45 i.e cost of ticket Is 275 and seats will be 550.