Question 389250
Each year a school’s booster club holds a dance to raise funds. In the past, the profit the club makes after paying for the bank and other costs has been modeled by the function P(t)= -16t² + 800t – 4000, where t represents the ticket price in dollars. 
a.	What ticket price gives the maximum profit?
b.	What is the maximum profit?
c.	What ticket price(s) would generate a profit of $5424? 


P(t) = -16t^2 + 800t - 4000
standard form parabola is y = ax^2 + bx + c,
here a = -16, b = 800, c = -4000,
vertex x-coordinate = -b/2a = -800/(2 * -16) = -800/-32 = 25
P(25) = -16 * 25^2 + 800 * 25 - 4000
P(25) = -16 * 625 + 20000 - 4000
P(25) = -10000 + 20000 - 4000
P(25) = 10000 - 4000 = 6000
$25 dollar ticket price gives maximum profit of $6000
for $5424 profit:
5424 = -16t^2 + 800t - 4000
0 = -16t^2 + 800t - 9424
0 = t^2 - 50t + 589
{{{t = (50 +- sqrt( (-50)^2 - 4*589 ))/2 }}}
{{{t = (50 +- sqrt( 2500 - 2356 ))/2 }}}
{{{t = (50 +- sqrt( 144 ))/2 }}}
{{{t = (50 +- 12)/2 }}}
t = 62/2 or t = 38/2
t = 31 or t = 19
either $19 or $31 dollar ticket prices