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The monthly output at the Olek Carpet Mill is Q(x) = 15,000 + 2x2 units, (40 ≤ x ≤ 60) where x is the number of workers employed at the mill. If there are currently 50 workers, find the instantaneous rate of change of monthly output with respect to the number of workers. That is, find Q'(50). Q'(50) =

Answer :

Answer:

[tex]Q'(50)=200[/tex]

Step-by-step explanation:

We have been given that the monthly output at the Olek Carpet Mill is [tex]Q(x) = 15,000+2x^2[/tex] units, [tex](40\leq x \leq60)[/tex] where x is the number of workers employed at the mill. We are asked to find instantaneous rate of change of monthly output with respect to the number of workers.

Let us find derivative of our given function.  

[tex]Q'(x) = \frac{d}{dx}(15,000)+\frac{d}{dx}(2x^2)[/tex]

[tex]Q'(x) = 0+4x[/tex]

[tex]Q'(x)=4x[/tex]

To find instantaneous rate of change at [tex]x=50[/tex], we will substitute [tex]x=50[/tex] in our derivative function as:

[tex]Q'(50)=4(50)[/tex]

[tex]Q'(50)=200[/tex]

Therefore, the instantaneous rate of change is 200 with respect to the number of workers.

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