The relationship between the average ticket price and the number of seats is linear, and so one way to find the maximum revenue that can be earned is to graph the function. If you substitute y for P and x for q, you can write the equation as y =...

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The relationship between the average ticket price and the number of seats is linear, and so one way to find the maximum revenue that can be earned is to graph the function. If you substitute y for P and x for q, you can write the equation as y = -1/5x + 22. Plotting this line, you get a y-intercept of (0, 22) and an x intercept of (110, 0). In essence, this means that if the ticket price was $22, no one would come, and if the price was $0, 110,000 fans would attend. Since the stadium seats 56,000, though, the point with the greatest fan to price ratio is (56, 10.8). In other words, at a price of $10.80 per ticket, 56,000 fans would buy tickets. 56 x 10.8 = 604.8 or $604,800.

The maximum is slightly higher, however, if Q = 55, because then P = 11. 55 x 11 = 605: a price of $11 will attract 55,000 fans and earn a revenue of $605,000. This answer can, again, be understood in terms of the graph. The region under the line in the first quadrant is a right triangle with sides of 22 and 110. The product of P and Q is equivalent to the area of a rectangle inscribed in the triangle. The greatest area of such a rectangle is obtained when the length and width are 1/2 as long as the legs of the triangle: 11 and 55. You can confirm this by plugging in 54 and 56 for Q: in both cases, you get a P that is slightly less than 605.