Question

A manufacturer has determined that the total cost C of operating a factory is
C = 2.5x2 + 75x + 25000,
where x is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is C/x.)
x = _____ units

Answer 1

Answer:

x = 100 units

Step-by-step explanation:

C = 2.5x^2 + 75x + 25000

To find average cost per unit, divide C by x:

C/x or Average cost (AC) per unit = $\frac{2.5x^2 + 75x + 25000}{x}$ = 2.5x + 75 + 25000/x ⇔ 2.5x + 75 + 25000x^-1 (equation 1)

To find cost minimising, we use differentiation (differentiate equation 1 in respect to x and set it equal to 0):

d(AC)/dx = 0

d(AC)/dx ⇔ 2.5 - 25000x^-2 = 0

2.5 = 25000x^-2

2.5 = $\frac{25000}{x^2}$

2.5x^2 = 25000

x^2 = 10000

x = $\sqrt{10000}$

x = 100 units

Answer 2

Cost functions are used to model the outputs from inputs.

The average cost per unit will be minimized at 100 units of production level.

The cost function is given as:

$\mathbf{C(x) = 2.5x^2 + 75x + 25000}$

Calculate the average cost function using

$\mathbf{A(x) = \frac{C(x)}{x}}$

So, we have:

$\mathbf{A(x) = \frac{2.5x^2 + 75x + 25000}{x}}$

Simplify

$\mathbf{A(x) = 2.5x + 75 + \frac{25000}{x}}$

Differentiate

$\mathbf{A'(x) = 2.5 + 0 - \frac{25000}{x^2}}$

$\mathbf{A'(x) = 2.5 - \frac{25000}{x^2}}$

Set to 0

$\mathbf{2.5 - \frac{25000}{x^2} = 0}$

Collect like terms

$\mathbf{-\frac{25000}{x^2} = -2.5}$

Cross multiply

$\mathbf{-2.5x^2 = -25000 }$

Make x^2 the subject

$\mathbf{x^2 = 10000 }$

Take square roots of both sides

$\mathbf{x = 100 }$

Hence, the average cost per unit will be minimized at 100 units of production level.

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