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]
