we know more than 110 units are produced the cost of producing x unit is given by c(x) = 0.20x³ - 25x² + 1531x + 25,000 how many units should be produced to have lowest possible average cost
average cost = c(x)/x = 0.20x^2 - 25x + 1531 + 25000/x
dc/dx = 0.40x - 25 - 25000/x^2
dc/dx=0 when x = 73.9
To determine the number of units that should be produced to have the lowest possible average cost, we need to find the minimum point of the average cost function.
The average cost is calculated by dividing the total cost by the number of units produced. In this case, the total cost is given by the function c(x) = 0.20x³ - 25x² + 1531x + 25,000, and the number of units produced is x.
The average cost function can be expressed as:
A(x) = c(x) / x
To find the lowest possible average cost, we need to find the minimum point of the average cost function. This can be done by finding the derivative of A(x) and setting it equal to zero.
First, let's find the derivative of A(x):
A'(x) = (c(x) / x)' = [(0.20x³ - 25x² + 1531x + 25,000)' / x] = (0.60x² - 50x + 1531) / x
Setting A'(x) equal to zero:
(0.60x² - 50x + 1531) / x = 0
To solve this equation, we can multiply both sides by x to get rid of the denominator:
0.60x² - 50x + 1531 = 0
This equation can be solved using algebraic methods (factoring, quadratic formula, etc.) or graphically (plotting the function and finding the x-coordinate of the minimum point).
Once you solve for x, the value obtained will represent the number of units that should be produced to achieve the lowest possible average cost.
To find the number of units that should be produced to have the lowest possible average cost, we need to analyze the given cost function, c(x) = 0.20x³ - 25x² + 1531x + 25,000.
The average cost is calculated by dividing the total cost by the number of units produced. In this case, the total cost is given by the cost function c(x), and we want to minimize the average cost.
To find the lowest possible average cost, we need to find the minimum value of the function c(x) and determine the corresponding value of x.
Here are the steps to find the number of units that should be produced:
1. Take the derivative of the cost function c(x) with respect to x. This will give us the rate of change of the cost with respect to the number of units produced.
2. Set the derivative equal to zero and solve for x. This will give us the critical points where the rate of change is zero.
3. Evaluate the second derivative of the cost function at the critical points. This will help us determine whether these points correspond to a minimum or maximum value.
4. Determine the minimum value of the cost function by plugging the critical points into the cost function and finding the corresponding cost.
5. Identify the corresponding number of units produced (x) for the minimum cost.
By following these steps, we can find the number of units that should be produced to have the lowest possible average cost.