A company's marginal cost function is [tex]C^{\prime}(x)=\frac{10}{\sqrt{x}}[/tex], where [tex]x[/tex] is the number of units.

Find the total cost function [tex]C(x)[/tex] and determine the cost of producing the first 100 units.

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

The total cost to produce 100 units is [tex]\square[/tex]. Select an answer: [tex]$\vee$[/tex]



Answer :

To solve for the total cost function [tex]\(C(x)\)[/tex] and the cost of producing the first 100 units, we need to integrate the marginal cost function. The marginal cost function given is:

[tex]\[ C'(x) = \frac{10}{\sqrt{x}} \][/tex]

1. Find the total cost function [tex]\(C(x)\)[/tex]:

To find [tex]\(C(x)\)[/tex], we integrate the marginal cost function with respect to [tex]\(x\)[/tex]:

[tex]\[ C(x) = \int \frac{10}{\sqrt{x}} \, dx \][/tex]

We can rewrite [tex]\(\frac{10}{\sqrt{x}}\)[/tex] as [tex]\(10x^{-\frac{1}{2}}\)[/tex]. Thus, we have:

[tex]\[ C(x) = \int 10x^{-\frac{1}{2}} \, dx \][/tex]

We now use the power rule of integration, which is [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\(n \neq -1\)[/tex]. Here, [tex]\(n = -\frac{1}{2}\)[/tex]:

[tex]\[ C(x) = 10 \int x^{-\frac{1}{2}} \, dx \][/tex]
[tex]\[ C(x) = 10 \left( \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} \right) \][/tex]
[tex]\[ C(x) = 10 \left( \frac{x^{\frac{1}{2}}}{\frac{1}{2}} \right) \][/tex]
[tex]\[ C(x) = 10 \cdot 2x^{\frac{1}{2}} \][/tex]
[tex]\[ C(x) = 20\sqrt{x} \][/tex]

Thus, the total cost function is:

[tex]\[ C(x) = 20\sqrt{x} \][/tex]

2. Determine the cost of producing the first 100 units:

To find the cost of producing the first 100 units, we substitute [tex]\(x = 100\)[/tex] into the total cost function [tex]\(C(x)\)[/tex]:

[tex]\[ C(100) = 20\sqrt{100} \][/tex]
[tex]\[ C(100) = 20 \cdot 10 \][/tex]
[tex]\[ C(100) = 200 \][/tex]

### Final Answers:
- The total cost function is [tex]\( C(x) = 20\sqrt{x} \)[/tex]
- The total cost to produce 100 units is 200