# Rate of Change

• April 13th 2010, 07:31 PM
akuczma86
Rate of Change
A company knows that unit cost C and unit revenue R from the production and sale of x units are related by $C= (R^2/176,000) + 9330$. Find the rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is$4000.
• April 14th 2010, 12:56 AM
tom@ballooncalculus
Related rates nearly always depend on the chain rule, so you might want to try filling up this pattern...

http://www.ballooncalculus.org/asy/diffChain/rates.png

... where straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x, because we're speaking of rates of change with respect to x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Here we're given a formula relating two variables, which is the top row...

http://www.ballooncalculus.org/asy/d...in/costRev.png

So differentiate with respect to the inner function, and the inner function with respect to x...

Spoiler:
http://www.ballooncalculus.org/asy/d...n/costReva.png

Clearly, you can now substitute the given values of R and dC/dx then solve for dR/dx.

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• April 14th 2010, 02:35 AM
HallsofIvy
Quote:

Originally Posted by akuczma86
A company knows that unit cost C and unit revenue R from the production and sale of x units are related by $C= (R^2/176,000) + 9330$. Find the rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is$4000.

$\frac{dC}{dt}= \frac{1}{98,000}R \frac{dR}{dt}$.

You are told that [tex]\frac{dC}{dR}= 12 and R= 4000. You can calculate C from $C= \frac{(4000)^2}{176000}+ 9330$ and put those numbers into the equation above.