Question

Revenue for the high school band concert is given by the function

y=-30x² + 250x, where x is the ticket price, in dollars. The cost of the
concert is given by the function y = 490 - 30x. At what ticket price will the
band make enough revenue to cover their costs?

Answer 1

between $2.34 and $7.00

Explanation:

Given revenue of y = -30x² +250x and costs of y = 490 -30x for ticket price x, you want to know the ticket price at which revenue covers cost.

Price

Since we want revenue greater than or equal to cost, we have ...

-30x² +250x ≥ 490 -30x

30x² -280x +490 ≤ 0 . . . . . . subtract the right-side expression

(3x -21)(3x -7)/3 ≤ 0 . . . . . . . . divide by 10, factor

(x -7)(3x -7) ≤ 0 . . . . . . . . . . . simplify

The zeros of this product are at x = 7/3 and x = 7. The product is negative for x-values between these zeros.

Costs are covered for ticket prices between 7/3 and 7 dollars. To ensure positive profit, the lower end of this range of prices should be set at $2.34, rather than $2.33.

Any ticket price between $2.34 and $7.00 will ensue the band makes enough to cover their costs.

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Additional comment

The attached graph shows the profit function. Costs are covered when profit is greater than or equal to zero (on or above the x-axis). The calculator shows the profit will be about $0.93 if the ticket price is $2.34. If it is $2.33, the band will lose about $0.47 on the engagement.

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