Suppose a company has fixed costs of $2400 and variable costs per unit of
3/4x + 1690 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is
1800 − 1/4x dollars per unit.
(a) Find the break-even points. (Enter your answers as a comma-separated list.)
x =
(b) Find the maximum revenue.
$
(c) Form the profit function P(x) from the cost and revenue functions.
P(x) =
Find the maximum profit.
$
(d) What price will maximize the profit? (Round your answer to the nearest cent.)
$
To find the break-even points, we need to determine the total revenue and total cost functions and then set them equal to each other.
The total revenue is the product of the selling price per unit and the number of units produced, which is given by (1800 - 1/4x)x.
The total cost is the sum of the fixed costs and the variable costs, which is given by 2400 + (3/4x + 1690)x.
(a) Setting the total revenue equal to the total cost will give us the break-even point(s):
(1800 - 1/4x)x = 2400 + (3/4x + 1690)x
To solve this equation, we can combine like terms and move all the terms to one side:
1800x - (1/4)x^2 = 2400 + 3/4x^2 + 1690x
Multiply everything by 4 to clear the fraction:
7200x - x^2 = 9600 + 3x^2 + 6760x
Rearrange the equation to get a quadratic equation:
4x^2 + 4456x - 9600 = 0
Now we can solve the quadratic equation for the break-even points. Use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
where a = 4, b = 4456, and c = -9600.
By plugging these values into the formula and solving for x, we will get the break-even points.
(b) The maximum revenue is obtained by maximizing the revenue function (1800 - 1/4x)x. This can be done by taking the derivative of the revenue function, setting it equal to zero, and solving for x. Then substitute the value of x into the revenue function to find the maximum revenue.
(c) The profit function P(x) is given by subtracting the total cost from the total revenue: P(x) = (1800 - 1/4x)x - [2400 + (3/4x + 1690)x]
(d) To find the maximum profit, we need to maximize the profit function P(x). This can be done by taking the derivative of the profit function, setting it equal to zero, and solving for x. Then substitute the value of x into the profit function to find the maximum profit.
After finding the maximum profit, we can then determine the price that will maximize the profit by substituting the value of x into the selling price equation (1800 - 1/4x).