PLease help me out and fully answer the question below as fast as you can! Thanks

Question

The profit, p(t), made at a fair depends on the price of the ticket, t. The profit is modelled by the function

p(t) = -37t^2+1776t-7500,

a) What is the maximum profit?

b) What is the ticket price of a ticket that gives the maximum profit?

c) What is the break even point?

Answer 1

take the derivative and set = 0

-74t + 1776 = 0

t = 1776/74 = $24 for a ticket

plug 24 into the quadratic equation

-37(24)^2 + 1776(24) - 7500 = 42624 - 21312 - 7500 = $13,812 = maximum profit

break even is when

-37t^2 + 1776t - 7500 = 0

use the quadratic formula, t = -b/2a + or - (1/2a)sqr(b^2 - 4ac) where 1776=b, a =-37, c=-7500

t = -1776/-74 + or - (1/74)sqr(1776^2 -4(37)(7500)) = 24 + or - (1/74)sqr(3,154,176-1,110,000)

= 24 + 1429.8/74 = 24 + 19.3 =- -$43.30 or 24-19.3 = $4.70