A company manufactures and sells a product. The cost of producing each unit is given by the marginal cost function [tex]C^{\prime}(x) = 8x^3 - 24x + 8[/tex], where [tex]x[/tex] is the number of units produced.

Find the total cost function [tex]C(x)[/tex] and determine the cost of producing 80 units.

1. Find the total cost function [tex]C(x)[/tex]:
[tex]C(x) = \int (8x^3 - 24x + 8) \, dx[/tex]

2. Determine the cost of producing 80 units.

[tex]C(80) = \square[/tex]

Round the answer to 2 decimal places as needed.



Answer :

To determine the total cost function [tex]\( C(x) \)[/tex] and the cost of producing 80 units, follow these steps:

1. Understand the Problem:
The marginal cost function [tex]\( C'(x) \)[/tex] represents the rate at which the cost changes with respect to the number of units produced, [tex]\(x\)[/tex]. The given marginal cost function is [tex]\( C'(x) = 8x^3 - 24x + 8 \)[/tex].

2. Find the Total Cost Function [tex]\( C(x) \)[/tex]:
To find the total cost function [tex]\( C(x) \)[/tex], we need to integrate the marginal cost function [tex]\( C'(x) \)[/tex]:

[tex]\[ C(x) = \int C'(x) \, dx = \int (8x^3 - 24x + 8) \, dx \][/tex]

3. Perform the Integration:
Integrate each term separately:

[tex]\[ C(x) = \int 8x^3 \, dx - \int 24x \, dx + \int 8 \, dx \][/tex]

Evaluate each integral:

[tex]\[ \int 8x^3 \, dx = 8 \cdot \frac{x^4}{4} = 2x^4 \][/tex]

[tex]\[ \int 24x \, dx = 24 \cdot \frac{x^2}{2} = 12x^2 \][/tex]

[tex]\[ \int 8 \, dx = 8x \][/tex]

Combine these results to obtain the total cost function:

[tex]\[ C(x) = 2x^4 - 12x^2 + 8x + C_0 \][/tex]

Here, [tex]\( C_0 \)[/tex] is the constant of integration. For simplicity, we will assume [tex]\( C_0 = 0 \)[/tex].

Hence, the total cost function is:

[tex]\[ C(x) = 2x^4 - 12x^2 + 8x \][/tex]

4. Determine the Cost of Producing 80 Units:
Evaluate the total cost function [tex]\( C(x) \)[/tex] at [tex]\( x = 80 \)[/tex]:

[tex]\[ C(80) = 2(80)^4 - 12(80)^2 + 8 \cdot 80 \][/tex]

Given the calculated result:

[tex]\[ C(80) = 81843840.0 \][/tex]

Thus, the total cost function is:

[tex]\[ C(x) = 2x^4 - 12x^2 + 8x \][/tex]

And the cost of producing 80 units is:

[tex]\[ \boxed{81843840.00} \text{ dollars} \][/tex]