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?
What is the range of ticket prices the concert promoters can charge and still make at least the minimum amount of money desired?
Answers
Steve
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
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
oops. Forgot to change the direction of the inequality when changing signs:
3p^2 - 140p + 1280 <= 0
3p^2 - 140p + 1280 <= 0