The price P per unit at which a company can sell all that it produces is given by the
function P(x) = 300 — 4x. The cost function is c(x) = 500 + 28x where x is the number
of units produced. Find x so that the profit is maximum.
profit = revenue - cost
revenue = demand * price
profit is thus x(300-4x) - (500+28x)
= -4x^2 + 272x - 500
This a parabola with vertex at (34,4124)
So, max profit is 4124 when x = 34
To find the value of x that maximizes profit, we need to first find the profit function. Profit is calculated by subtracting the cost function from the revenue function.
The revenue function is given by P(x) = 300 - 4x, which represents the price per unit multiplied by the number of units sold.
The cost function is given by c(x) = 500 + 28x, which represents the fixed cost plus the variable cost per unit multiplied by the number of units produced.
So, the profit function can be calculated as follows:
Profit(x) = Revenue - Cost
Profit(x) = P(x) * x - c(x)
Profit(x) = (300 - 4x) * x - (500 + 28x)
To maximize the profit, we need to find the value of x that maximizes the profit function. To do this, we can take the derivative of the profit function with respect to x and set it equal to zero.
d(Profit(x))/dx = 0
d/dx((300 - 4x) * x - (500 + 28x)) = 0
To solve this equation, we can expand and simplify the equation to find x:
300x - 4x^2 - 500 - 28x = 0
-4x^2 + 272x - 500 = 0
We can solve this quadratic equation by factoring, completing the square, or applying the quadratic formula.
Once we obtain the value(s) of x, we can substitute it back into the profit function to find the maximum profit.