Here's another one that I do not really understand.
A company estimates that the cost (in dollars) of producing x units of a certain product isgiven by: C=800 + .04x + .0002x^2. Find the production level that minimizes the average cost per unit.
You have a typo. There is no minimum of the function you typed for positive x
dC/dx = 0 at max or min
dC/dx = 0 = .04 + .0004 x
revenueR(x)=10(x) and cost c(x) =2.5x + 1200. how many units must be produced each month to breakeven?
To find the production level that minimizes the average cost per unit, we need to find the minimum point of the average cost function.
The average cost is given by the formula: Average Cost (AC) = Total Cost (C) / Number of Units (x).
First, let's express the total cost as a function of the number of units produced (x).
The cost function is given as: C = 800 + 0.04x + 0.0002x^2.
Next, let's find the average cost function.
Average Cost (AC) = Total Cost (C) / Number of Units (x)
AC = (800 + 0.04x + 0.0002x^2) / x
Now, we need to find the minimum of the average cost function by taking the derivative and setting it equal to zero.
To do this, differentiate AC with respect to x:
d(AC)/dx = (800 + 0.04x + 0.0002x^2)' / x
Using the power rule and chain rule, we can simplify this:
d(AC)/dx = (0.04 + 0.0004x) / x
Setting this derivative equal to zero gives:
0.04 + 0.0004x = 0
Solving for x, we get:
0.0004x = -0.04
x = -0.04 / 0.0004
x = -100
However, since we cannot have a negative number of units produced, we discard this value.
Therefore, there is no production level that minimizes the average cost per unit in the given situation.
To find the production level that minimizes the average cost per unit, we need to differentiate the cost function with respect to x and set the derivative equal to zero. Here's a step-by-step explanation on how to do it:
1. Start with the cost function: C = 800 + 0.04x + 0.0002x^2
2. Calculate the average cost per unit by dividing the cost by the number of units produced: AC = C/x
3. Substitute the cost function into the average cost equation: AC = (800 + 0.04x + 0.0002x^2)/x
4. Simplify the expression by dividing each term by x: AC = 800/x + 0.04 + 0.0002x
5. To minimize the average cost, we need to find the value of x that makes the derivative of the average cost function equal to zero.
6. Take the derivative of the average cost function with respect to x: d(AC)/dx = -800/x^2 + 0.0002
7. Set the derivative equal to zero and solve for x: -800/x^2 + 0.0002 = 0
8. Rearrange the equation to isolate x: -800/x^2 = -0.0002
9. Multiply both sides by x^2: -800 = -0.0002x^2
10. Divide both sides by -0.0002: 4000000 = x^2
11. Take the square root of both sides: x = √4000000
12. Simplify: x = 2000
Therefore, the production level that minimizes the average cost per unit is 2000 units.