# maximizing income and quadratic function

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?

• Yes this seems correct. – Peter Foreman Aug 17 '19 at 23:49

$$f(t)\,=(50\,+5t)\,(1000-\,10t)$$ 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.