I'm not even sure what kind of math this is. lol I'm posting this for my son who is in prison.

S

SusanCoutu

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Sep 28, 2018
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Hi all, I have a problem I can't figure out. It goes like this:



A product sells for $3.50 currently and has a demand function of

p = 8000/q

Suppose manufacturing costs are increasing at a rate of 15% over time and the company plans to increase the price p at this rate as well. Find the rate of change of demand over time.


But when I differentiate this problem for time, I get:

dq/dt = -8000 * p^(-2) * dp/dt


Assuming p, the price, equals $3.50, and the rate of change of the price, dp/dt, equals 1.15 (15% increase), the answer I get is approximately -751, when my textbook is telling me it should be -343. Where am I going wrong?
 
We have to determine what it means for costs to "increase at a rate of 15% over time". As I read it, that would not be a linear increase (because it is 15% of a changing quantity, not just a constant 15% of some initial amount), but an exponential increase like compound interest.

In particular, the rate of change of p is not 1.15; that is what it is multiplied by each time period (in some sense), not what is added to it.

But if we suppose a continuously compounded amount, rather than literally increasing by 15% each year, we can say that dp/dt = 0.15p (that is, the rate of increase is 15% of the current price. And this gives their answer (try it!), so maybe it's what they meant.
 
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