If the total cost function for producing x lamps is

C(x) = 90 + 36x + 0.1x^2
dollars, producing how many units will result in a minimum average cost per unit?
PLEASE EXPLAIN YOUR WORK
Thank you

avg cost is c(x)/x = .1x + 36 + 90/x

now just find x when dc/dx = 0.

To find the minimum average cost per unit, we need to find the derivative of the total cost function and set it equal to zero.

Step 1: Find the derivative of the total cost function, C(x), with respect to x.
C'(x) = 36 + 0.2x

Step 2: Set the derivative equal to zero and solve for x.
36 + 0.2x = 0
0.2x = -36
x = -36 / 0.2
x = -180

Step 3: Analyze the result.
The solution x = -180 is not applicable in this scenario because it represents a negative number of lamps, which is not possible in this context since it is not physically realistic.

Therefore, we need to apply another method. In this case, we will consider the vertex of the parabola formed by the cost function.

Step 4: Determine the x-coordinate of the vertex using the formula:
x = -b / (2a)

From the given cost function C(x) = 90 + 36x + 0.1x^2, we can identify that a = 0.1 and b = 36.

x = -36 / (2 * 0.1)
x = -36 / 0.2
x = -180

Step 5: Analyze the result.
We again find that x = -180. However, we can see that the vertex corresponds to a positive value of x (the number of lamps), which makes it physically realistic.

So, producing x = 180 lamps will result in a minimum average cost per unit.

To find the production level that results in a minimum average cost per unit, we need to determine the production level at which the derivative of the average cost function is equal to zero.

The average cost per unit is given by the equation:

AC(x) = C(x)/x

Where C(x) is the total cost function and x is the number of lamps produced.

To find the derivative of the average cost function, we need to differentiate C(x) with respect to x and divide it by x:

AC'(x) = [C'(x) * x - C(x)] / x^2

First, let's find C'(x), the derivative of the total cost function:

C'(x) = d/dx (90 + 36x + 0.1x^2)
= 36 + 0.2x

Next, substitute C'(x) into the average cost derivative equation:

AC'(x) = [(36 + 0.2x) * x - (90 + 36x + 0.1^2)] / x^2
= (36x + 0.2x^2 - 90 - 36x - 0.1x^2) / x^2
= (0.1x^2 - 54) / x^2
= 0.1 - (54/x^2)

To find the production level that results in minimum average cost per unit, we set AC'(x) equal to zero and solve for x:

0.1 - (54/x^2) = 0

0.1 = 54/x^2

x^2 = 54/0.1

x^2 = 540

Taking the square root of both sides:

x = √540

Simplifying this radical:

x ≈ 23.2379

Therefore, producing approximately 23 lamps will result in minimum average cost per unit.