We can write that as a function:
[tex]f(n)=(200 -20n) * (300 + 50n)\\f(n)=60000+10000n-6000n-1000n^2\\f(n)=-1000n^2+4000n+60000\\[/tex]
where [tex]n[/tex] is the number of price reductions. We can see that it is a quadratic function which has the biggest value at P = (p, q). It means that p is the number of reductions which results in the biggest profit.
[tex]p=\frac{-b}{2a}=\frac{-4000}{-2000}=2[/tex]
It means that it's the best to make 2 price reductions for the company. The tickets should be sold for $200 - 2 * $20 = $160. It results in 2 * 50 more tickets sold which is 300 + 100 = 400.
Answer:
Ticket price: $160
Number of tickets: 400
