How can I do this quadratic question? Thanks

Question

A concert is soon holding and the venue can hold 12000 people. The minimum price is $100 and all tickets can be sold out if the price of each ticket is set to this minimum. For very increment of $1 in the price, the number of tickets sold decreases by 20. It is given that the total expenditure of the performance is $1000000. Let $x be the price of each ticket and $y be the overall profit of the concert.

(a) Show that the number of tickets sold at $x is 14000-20x.

(b) Show that y=-20x^2 +14000x -1000000

(c) write down the domains of the function in (a)

(d) Find the range of prices of each ticket such that there is an overall profit.

(e) Find the price of each ticket such that the profit is at maximum and the number of ticket sold at this price.

Answer 1

First you are going to want to try to make sense of the ideas in this word problem, which is not very easy to get into at first. Think about how you can word it so that it makes a little more sense, then answer my follow-up questions:

1. There's going to be a concert and 12000 people are going to show up if the tickets cost $100 each. Follow-up- how much money would they get from ticket sales at this price?
2. Every time you raise the price by $1, 20 people decide to skip the concert because it's too expensive. So if your price is $101 per ticket then 11980 people will come, at $102 per ticket, 11960 will come, and so on. Follow-up- how does this relate to (a)?
3. The reason that this ends up being a quadratic is that as you raise the price, you get more money per ticket but fewer people want to come. The concert will bring in money equal to ticket price multiplied by number of tickets sold. Follow-up- we already know that the ticket sale equation was given in (a). We know that ticket price is just x. Now think about the idea of profit, which is y. Profit is just the money you bring in minus the cost of doing the event. Here the cost is a million dollars (1000000). Use (tickets sold equation)*(ticket price) - (event cost) to get (b).
4. Remember that domain is related to x values that are possible. Here there is a minimum which the problem tells you, so that is going to be the minimum of our domain. You will have to find the maximum by figuring out the price that you can set where nobody will buy a ticket. Any x value beyond that won't make sense because the equation would tell you that there are a negative number of ticket sales. Just focus on the equation from (a), which is ticket sales, and find when it would equal zero. That is going to be the max of your domain.
5. For (d) you are looking for the range of ticket prices that are going to bring in more than the million dollar cost of the event. You want to find the zeroes, or x-intercepts, of your parabola, to find your range. Keep in mind, though that you have to ignore any part that is outside of the domain you found in (c).
6. Finally, you want the biggest profit possible. This is going to be the highest point in the graph of the equation, which is a quadratic. This is the vertex. Remember that you find the x-coordinate of the vertex by looking at the standard form (ax² + bx + c) and solving -b/2a.

I hope this helps. Good luck!