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This week, a factory is producing 50 units of a particular commodity, and the amount being produced is increasing at the rate of two units per week. If $C(x)$ is the total cost of producing x units, and $C(x) = 0.08x^3-x^2+10x+48$, find the current rate at which the production cost is increasing.

How I approach the problem:

$C^\prime(x) =0.08(3)x^2-2x+10$
$C^\prime(50) = \$510$

So the production cost is increasing at a rate of $\$510$?? That seems unreasonable; furthermore, what is to be done with "and the amount being produced is increasing at the rate of two units per week" or is that outside info, since the problem is asking for the rate of change in production at $50$ units?

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First, rewrite it into maths!

Let $P(t)$ be the number of units being produced, where $t$ is the time (measured in weeks) from the start of the process (this week).

Then we know $$\frac{dP}{dt}=2.$$

Now $C(x)$, the cost for producing $x$ units in a week becomes $C(P(t))$, the cost of producing $P(t)$ units in a week, so $$C(P(t))=0.08P(t)^3-2P(t)+10.$$ Now the question is to find $$\frac{dC}{dt}.$$

Does that give you an idea about how to start?

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By chain rule, $\frac{dC} {dt} = \frac{dC} {dx} \cdot \frac{dx} {dt} $

You are given $\frac{dx} {dt} = 2 $ (per week) with the instantaneous value of $x$ at that point in time being $50$.

So first find $C'(x) = \frac{dC} {dx} = 0.24x^2 - 2x + 10$ and then $C'(50) = 510$.

That gives (at that point in time) $\frac{dC} {dt}\mid_{x=50} =C'(50) \cdot 2 = 1020 $ and that's your answer (in appropriate monetary rate units e.g. dollars per week).

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