Rosalie is organizing a circus performance to raise money for a charity. She is trying to decide how much to charge for tickets. From past experience, she knows that the number of people who will attend is a linear function of the price per ticket. If she charges 5 dollars, 1160 people will attend. If she charges 7 dollars, 930 people will attend. How much should she charge per ticket to make the most money

  $7.54

Step-by-step explanation:

Using the two-point form of the equation of a line, we can write the equation for the number tickets (t) as a function of price (p).

  t = (t2 -t1)/(p2 -p1)(p -p1) +t1

  t = (930 -1160)/(7 -5)(p -5) +1160

  t = -230/2(p -5) +1160

  t = -115p +1735 = -115(p -15 2/23)

The revenue from ticket sales will be the product of the price and the number of tickets sold:

  r = pt = p(-115)(p -15 2/23)

This is the equation of a downward-opening parabola with zeros at p=0 and p=15 2/23. The vertex of the parabola (maximum revenue) will be found at a ticket price halfway between these values. The price for maximum revenue is ...

  (0 +15 2/23)/2 = 7 25/46 ≈ 7.54

Rosalie should charge $7.54 per ticket to obtain the most revenue.

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