Question

Kevin operates a movie theater that sells about 800 tickets per day for $8 each. Kevin predicts that for each $0.50 increase in the ticket price, 40 fewer tickets will be sold. Let x represent the number of $0.50 price increases and f(x) represent the total earnings from ticket sales.

f(x)=-20x^2+80x+6400


Kevin wants to determine the best potential earnings from ticket sales each day.


Complete the following statement for this situation.


The function reveals that the (minimum or maximum) earnings from ticket sales will be (320,6400,6480 or 720$) after(6,8,4 or 2) $0.50 price increases.

Answer 1

maximum, 6480, two

Answer 2

The function reveals that the maximum earning from ticket sales will be $6480 after 2 ($0.50) price increases

Explanation:

* Lets explain how to solve the problem

- The function f(x) = -20x² + 80x + 6400 represents the total

earnings from ticket sales and x represents the number of $0.50

price increases

∵ f(x) is a quadratic function

∵ The coefficient of x² is -20

∴ The function has a maximum value

- The vertex of the quadratic function f(x) = ax² + bx + c is (h , k),

where h = -b/2a and k = f(h)

* Lets find the maximum point of f(x)

∵ a = -20 , b = 80

∵ h = -b/2a

∴ h = -(80)/2(-20) = -80/-40 = 2

∵ k = f(h) and h = 2

∴ f(2) = -20(2)² + 80 (2) + 6400

∴ f(2) = -80 + 160 + 6400

∴ f(2) = 6480

∴ k = 6480

* The function reveals that the maximum earning from ticket sales

will be $6480 after 2 ($0.50) price increases