# mylesbibbs' question at Yahoo! Answers regarding marginal cost

#### MarkFL

Staff member
Here is the question:

CALC. HELP!!!!!! Related Problems!!?!!?

The weekly cost C, in dollars, for a manufacturer to produce q automobile tires is given by
C = 2300 + 15q − 0.01q2 0 ≤ q ≤ 800.
If 400 tires are currently being made per week but production levels are increasing at a rate of 30 tires/week, compute the rate of change of cost with respect to time.
Here is a link to the question:

CALC. HELP!!!!!! Related Problems!!?!!? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

#### MarkFL

Staff member
Hello mylesbibbs,

We are given:

$$\displaystyle C(q)=2300+15q−0.01q^2$$

and we are asked to find $$\displaystyle \frac{dC}{dt}$$

Now, if we take the given cost function and differentiate with respect to time, using the chain rule we find:

$$\displaystyle \frac{dC}{dt}=\frac{dC}{dq}\cdot\frac{dq}{dt}= \left(15-0.02q \right)\frac{dq}{dt}$$

We are also told that $$\displaystyle \frac{dq}{dt}=30$$ and so we have:

$$\displaystyle \frac{dC}{dt}=30(15-0.02q)$$

and so at the current production level of $q=400$, we find:

$$\displaystyle \left.\frac{dC}{dt}\right|_{q=400}=30(15-0.02\cdot400)=210$$

Thus, we find that the production cost is increasing at a rate of \\$210 per week.

To mylesbibbs and any other guests viewing this topic, I invite and encourage you to post other marginal cost questions here in our Calculus forum.

Best Regards,

Mark.