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
[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
