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 - x^2 - 2000 - 50x

p(x) = - x^2 + 780X - 2000

to determine the maximum profit of the company we need to find the maximum of the function p(x)

p(x) = - x^2 + 780X - 2000 is quadratic function

since the quotient in front of x^2 is -1 < 0 the function has maximum in the vertex

for the function

f(x) = ax^2 + bx + c

the maximum value = c^2 - b^2/4a

in our case

a = -1

b = 180

c = - 2000

max p(x) = (-2000)^2 - 180^2/4*(-1) = $4,008,100

The maximum profit is $4,008,100.