Question

The marginal cost of producing the xth box of light bulbs is LaTeX: 10-\frac{x}{10000},\:\:\:\:\:\:10 − x 10000 ,and the fixed cost is $15,000. Find the cost function LaTeX: C\left(x\right)C ( x ).

Answer 1

To properly answer this question, let us first define what marginal cost is:

Marginal cost can be defined or refers to the rate of change of the total cost.

Referencing back to the question on how to get the fixed cost , we can do this by integrating the marginal cost function. Also, the fixed cost is given in the question which is the total cost when the production is zero. By using this, we can find the value of integration constant which will appear in the total cost function

Therefore,

∫ xⁿdx ****= xⁿ⁺¹/n+1

M C = 10 - x / 10000

dc/dx = 10 - x / 10000

At this point, we try to separate the variables:

dc = [ 10 - x / 10000 ] dx

After the separation, we perform the integration on both sides:

∫ dc = ∫ [ 10 - x / 10000 ] dx

Recalling that K here is the constant of integration, we then have:

c = 10x - x² / 15000 + k

Also, referencing back to the question, we note that fixed cost is the total cost when the production level is zero, that is C = 15000 , when x = 0 .

Therefore:

15000 = 10 (0) - (0)² / 15000 + k

15000 = 0 + k

k = 15000

c = 10x - x² / 15000 + k

c = 10x - x² / 15000 + 15000

Therefore, the total cost function is:

C = 10 x - x² / 15000 + 15000

So, in summary, the cost of function can be said to be C = 10 x - x² / 15000 + 15000

Answer 2

`C(x)= 10x-(x^2)/(20000) + 15000`

Explanation:

Cost function (C(x)) = Fixed cost (F) + Variable cost (V(x))

Marginal cost (M(x)) = slope of Variable cost (V(x))

`V(x)=\int M(x) dx`

`V(x)=\int 10-(x)/(10000)dx`

`V(x)= 10x-(x^2)/(20000)`

Note that we do not consider the constant of integration. This constant is the fixed cost.

`C(x)= V(x) + 15000`

`C(x)= 10x-(x^2)/(20000) + 15000`