Question

A concert is selling tickets. Their last concert sold 1000 tickets for $20 each. Market research

suggests that for every $1 increase in price, 90 fewer people would come. By how much should
they increase (or decrease) ticket prices?

ie: use the factoring methods if possible (common factoring, simple
factoring, decomposition factoring), then use completing the square/quadratic formula if an
equation is unfactorable).

Answer 1

Increase price by $44.44 per ticket for maximum concert revenue.

Method 1: Factoring (Simple Factoring):

1. Define variables:

- x = increase in price per ticket ($).

- y = number of tickets sold.

2. Revenue function:

- Revenue = y * (20 + x).

3. Market research equation:

- y = 1000 - 90x.

4. Substitute y in the revenue function:

- Revenue = (1000 - 90x) * (20 + x)

- Expand and factor:

- Revenue = 20000 - 8000x - 90x^2

- Revenue = -90(x^2 + 88.88x + 222.22)

5. Analyze the factored term:

- x^2 + 88.88x + 222.22 = (x + 44.44)^2 + 7.4044

- This is a minimum point for the parabola, meaning the revenue is maximized at this point.

Therefore, the ticket price should increase by x = $44.44.

Method 2: Completing the Square (Quadratic Formula):

1. Follow steps 1-3 from Method 1.

2. Set the derivative of the revenue function equal to 0:

- Revenue' = 20000 - 16000x - 180x^2 = 0

3. Solve for x using the quadratic formula:

- x = (-b ± √(b^2 - 4ac)) / 2a

- x = (16000 ± √(16000^2 - 4 * -180 * 20000)) / 2 * -180

- x = 44.44

Again, the ticket price should increase by x = $44.44.