suppose (c(x)=0.03x^{2}+3x + 3600) is the total cost of a company to produce (x) units of a certain product…

suppose (c(x)=0.03x^{2}+3x + 3600) is the total cost of a company to produce (x) units of a certain product. find the production level (x) that minimizes the average cost (a(x)=\frac{c(x)}{x}). (round the production level to two decimal places.)

suppose (c(x)=0.03x^{2}+3x + 3600) is the total cost of a company to produce (x) units of a certain product. find the production level (x) that minimizes the average cost (a(x)=\frac{c(x)}{x}). (round the production level to two decimal places.)

Answer

Explanation:

Step1: Find the average - cost function

Given $C(x)=0.03x^{2}+3x + 3600$, then $A(x)=\frac{C(x)}{x}=0.03x + 3+\frac{3600}{x}$.

Step2: Differentiate the average - cost function

The derivative of $A(x)$ with respect to $x$ is $A^\prime(x)=0.03-\frac{3600}{x^{2}}$.

Step3: Set the derivative equal to zero

Set $A^\prime(x) = 0$, so $0.03-\frac{3600}{x^{2}}=0$. Then $\frac{3600}{x^{2}}=0.03$. Cross - multiply to get $0.03x^{2}=3600$.

Step4: Solve for $x$

Divide both sides of $0.03x^{2}=3600$ by $0.03$: $x^{2}=\frac{3600}{0.03}=120000$. Take the square root of both sides. Since $x$ represents a production level (non - negative), $x=\sqrt{120000}\approx346.41$.

Answer:

$x\approx346.41$