Word problem, quadratic equations

Question

The valley highschool student counsil is sponsoring a dance. They anticipate that 500 students will attend. The ticket price is $3.00, and they estimate that for every $0.20 increase in ticket price, 25 fewer students will attend. What ticket price will maximize the student concil's profit

Answer 1

The problem is not concisely given. It seems to assume 500 students will attend which makes the problem ill defined. If we assume the number of students can vary then let x be the number of 0.20 increments in cost and the number of 25 student increments then the cost is (3+.2x)(500-25x) or -5x^2+25x+1500. We can differentiate to find the maximum or -10x+25=0 or the number of 0.2 increments is 2.5. and the cost is 3.50 and the number of students is 500-62.5=437.5 and the maximum profit is 437.5*3.5 = 1531.25.