If the total cost of manufacturing a commodity is $C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + 4 x + 200$ dollars when x units are produced, for what value of x is the average cost the least?
A) 37
B) 38
C) 39
D) 40

Question

Quizplus · Accepted Answer

The average cost $A(x)$ is given by $A(x) = \frac{C(x)}{x} = \frac{1}{8}x + 4 + \frac{200}{x}$. To find the minimum average cost, we differentiate $A(x)$ with respect to $x$ and set the derivative equal to zero. The derivative of $A(x)$ is $A'(x) = \frac{1}{8} - \frac{200}{x^2}$. Setting $A'(x) = 0$ gives $\frac{1}{8} = \frac{200}{x^2}$, which simplifies to $x^2 = 1600$. Taking the positive square root gives $x = 40$, which is the value of $x$ that minimizes the average cost.

If the Total Cost of Manufacturing a Commodity Is C(x)=18x2+4x+200C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + 4 x + 200C(x)=81​x2+4x+200

If the Total Cost of Manufacturing a Commodity Is $C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + 4 x + 200$