the profit of a company, in dollars, is the difference between the company's revenue and cost. the cost C(x), and R(x) are functions for a particular company. the x represents the number of items produced and sold to distributors. C(x)=2000+60x R(x)=860x-x^2

Determine the maximum profit of the coming.

p = R - C = -x^2 + 800 x - 2000

dp/dx = -2x + 800 = 0 at max or min
so x = 400 at max or min of p
is it a max or a min
well if x is large - or +, the sides of the parabola zoom negative so the vertex was a max (if you do not know calculus complete the square to find vertex of that parabola)
then
p = -160,000 + 800(400) - 2000
= -160, 000 + 320,000 - 2,000

how do you find dp/dx?

Ah, calculating profits, eh? I'm here to clown around with numbers! To find the maximum profit, we need to find the peak of the profit function. And guess what? The profit is the difference between revenue and cost!

So, let's find the profit function P(x). We have C(x) = 2000 + 60x as the cost function, and R(x) = 860x - x^2 as the revenue function. Subtracting C(x) from R(x), we get:

P(x) = R(x) - C(x)
P(x) = (860x - x^2) - (2000 + 60x)
P(x) = 860x - x^2 - 2000 - 60x
P(x) = -x^2 + 800x - 2000

Now, to determine the maximum profit, we need to find the vertex of the parabola described by the profit function. The x-coordinate of the vertex is given by:

x = -b / (2a)

Here, a = -1 and b = 800. Plugging these values into the equation, we get:

x = -800 / (2 * -1)
x = -800 / -2
x = 400

So, the maximum profit occurs when x = 400. Now, let's find the corresponding y-coordinate (profit):

P(400) = -400^2 + 800 * 400 - 2000
P(400) = -160000 + 320000 - 2000
P(400) = 158000

Therefore, the maximum profit for the company is $158,000. Hats off to that! 🎩

To determine the maximum profit of the company, we need to find the maximum point of the profit function. The profit function can be calculated by subtracting the cost function from the revenue function.

Given:
Cost function: C(x) = 2000 + 60x
Revenue function: R(x) = 860x - x^2

To find the profit function, we subtract the cost function from the revenue function:
P(x) = R(x) - C(x)
= (860x - x^2) - (2000 + 60x)
= 860x - x^2 - 2000 - 60x
= -x^2 + 800x - 2000

The maximum profit can be determined by finding the vertex of the quadratic function, since the coefficient of the x^2 term is negative.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a)

In our case, a = -1 (coefficient of x^2 term) and b = 800 (coefficient of x term).

x = -800 / (2 * -1)
x = 400

Now, substitute the value of x back into the profit function to find the maximum profit:

P(x) = -x^2 + 800x - 2000
P(400) = -(400)^2 + 800(400) - 2000
P(400) = -160000 + 320000 - 2000
P(400) = 158000

Therefore, the maximum profit of the company is $158,000.