800 people will attend a concert if tickets cost $20 each. Attendance will decrease by 30 people for each $1 increase in the price. The concert promoters need to make a minimum of $12 800.

Question

800 people will attend a concert if tickets cost $20 each. Attendance will decrease by 30 people for each $1 increase in the price. The concert promoters need to make a minimum of $12 800.
What is the range of ticket prices the concert promoters can charge and still make at least the minimum amount of money desired?

Steve · Answer

attendance at price p is
a = 800 - 30(p-20) = 1400-30p
we need
p*a >= 12800
p(1400-30p) >= 12800
1400p - 30p^2 - 12800 >= 0
3p^2 - 140p + 1280 >= 0
p <= 2/3 (35+√265)
p <= 34.19

So, as long as you charge less than $35, income will be ok.

Check:
at p=34 a=380 so sales=12,920
at p=35 a=350 so sales=12,250

Steve · Answer

oops. Forgot to change the direction of the inequality when changing signs: 3p^2 - 140p + 1280 <= 0