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A manufacturer has determined that the total cost C of operating a factory is

A manufacturer has determined that the total cost C of operating a factory is-example-1
User Skunkwerk
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1 Answer

1 vote

We have the next cost given function:


C=2.5x^2+75x+25000

The average cost function is given by:


A(x)=(C(x))/(x)

Replace:


A(x)=(2.5x^2+75x+25000)/(x)

Simplify the expression:


\begin{gathered} A(x)=(2.5x^2)/(x)+(75x)/(x)+(25000)/(x) \\ \text{Then:} \\ A(x)=2.5x+75+(25000)/(x) \end{gathered}

Derivate A(x)


A^(\prime)(x)=2.5(1)+0-(25000)/(x^2)

Set A'(x)=0


0=2.5(1)+0-(25000)/(x^2)

Solve for x:


\begin{gathered} 0=2.5-(25000)/(x^2) \\ -2.5=-(25000)/(x^2) \\ x^2=(-25000)/(-2.5) \\ x^2=10000 \\ \text{Take square root of both sides:} \\ \sqrt[]{x^2}=\sqrt[]{10000} \\ \text{Hence } \\ x=100 \end{gathered}

Therefore, the average cost per unit will be minimized at 100 units of production level.

User Leebriggs
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