4. let ( c(x) ) be the cost of producing ( x ) units of a commodity. if\n( c(x)=0.3 x^{2}+2 x )\nand the…

Explanation:

Step1: Integrate the marginal cost function

We know that (C(x)=\int C^{\prime}(x)dx). Given (C^{\prime}(x) = 0.3x^{2}+2x), then [ \begin{align*} C(x)&=\int(0.3x^{2}+2x)dx\ &=0.3\int x^{2}dx + 2\int xdx\ \end{align*} ] Using the power - rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (0.3\times\frac{x^{3}}{3}+2\times\frac{x^{2}}{2}+K), which simplifies to (0.1x^{3}+x^{2}+K).

Step2: Determine the constant of integration

The fixed cost is (C(0)). When (x = 0), (C(0)=2000). Substituting (x = 0) into (C(x)=0.1x^{3}+x^{2}+K), we get (C(0)=K). So (K = 2000). Then (C(x)=0.1x^{3}+x^{2}+2000).

Step3: Calculate the cost of producing 20 units

Substitute (x = 20) into (C(x)): [ \begin{align*} C(20)&=0.1\times(20)^{3}+(20)^{2}+2000\ &=0.1\times8000 + 400+2000\ &=800+400 + 2000\ &=3200 \end{align*} ]

Answer:

(C(x)=0.1x^{3}+x^{2}+2000) and the cost of producing (20) units is ($3200)