Answer:
maximum profit = $1,170,400
Step-by-step explanation:
let
p(x) be the profit as a function of x (x - the number of items produced and sold)
the profit of a company, in dollars, is the difference between the company's revenue and cost
p(x) = r(x) - c(x)
p(x) = 830x - x² - 2300 - 50x
p(x) = - x²+ 780X - 2300
to determine the maximum profit of the company we need to find the maximum of the function p(x)
p(x) = - x²+ 780X - 2300 is quadratic function
since the quotient in front of x² is -1 < 0 the function has maximum in the vertex
for the function
f(x) = ax² + bx + c
the maximum value = c² - b² ÷ 4 a
in our case
a = -1
b = 780
c = - 2300
max p(x) = (-2300)² - (780)² ÷(4× (-1))
= (5290000 - 608400 ) ÷ (-4) =$ 1,170,400
The maximum profit is $1,170,400