The planning committee for the upcoming school play “Miss-terious” at LMSA asked the mathematics classes to give
them some estimates about income that could be expected at different ticket price levels. The class did some market
research to see what students would be willing to pay for tickets. They reported back the following model: I = -75p2
+
600p, where I stands for income and p for ticket price, both in dollars.
a. Find the predicted income if ticket prices are set at $3.
b. Write equations that can be used to help answer each of the following questions. Then solve those equations,
check your solutions, and explain how you found the solutions.
i. What ticket price will give income of $1,125?
ii. What ticket price will give income of $900?
iii. What ticket price will give income of $970?
c. Find the price that will give maximum income, then find the maximum income.
i = -75 p^2 + 600 p
if p = 3
i = -75 (9) + 1800
1,125 = -75 p^2 + 600 p
75 p^2 -600 p + 1,125 = 0
solve quadratic for 2 values of p. Throw out any negative one
etc
for max
I assume you do not know calculus
so complete square to find vertex
75 p^2 - 600 p = -i
p^2 - 8 p = -i/75
p^2 - 8 p + 16 = -i/75 + 16
(p-4)^2 = -(1/75)(i-1200)
at vertex price for max is $4 and income is $1200