Expert: Paul Klarreich - 6/21/2006

Question
Recall that revenue is defined R=pq. Suppose you work for a
company that has revenue this month of $27,000 with a sales rate
of  q= 9000 units of their main product. Suppose that in addition
to an ongoing price increase of $0.12 a month, your company’s
sales are increasing at a rate of 30 units a month. Your boss asks
you to write a report describing how fast is the company’s revenue
is increasing. Set this up as a related rates problem (i.e., no credit
for just winging the solution), expressing all the relevant quantities
in terms of the given variables and their derivatives (with respect to
time   in months).  

Answer
Hi, Todd,

Subject:  Related Rates Revenue
Question:  Recall that revenue is defined R=pq. Suppose you work for a company that has revenue this month of $27,000 with a sales rate of  q= 9000 units of their main product. Suppose that in addition to an ongoing price increase of $0.12 a month, your company’s sales are increasing at a rate of 30 units a month. Your boss asks you to write a report describing how fast is the company’s revenue
is increasing. Set this up as a related rates problem (i.e., no credit for just winging the solution), expressing all the relevant quantities in terms of the given variables and their derivatives (with respect to time   in months).  

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Is this your first attempt at R-R problems?  If so, the scheme is something like this:

1. Identify the variables in the problem -- the things that change.  Give them names.
2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.
3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, your general knowledge and brilliance, whatever you have to.  This is the key step.
4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.

1. The variables are:
p = price per unit?
q = monthly sales.
R = monthly revenue

2. The rates of change are:

dp/dt = price change per month = 0.12
dq/dt = increase of sales per month = 30
dR/dt = increase in revenue, TO BE FOUND.

3. The relation is

R = pq.

4. Differentiate: (product rule, of course)

dR/dt = p dq/dt + q dp/dt

Now put in the appropriate numbers (your text doesn't make this totally clear.) which could be, that in month zero,  q = 9000, p = 3.