Expert: Paul Klarreich - 11/23/2005

Question
Dear Mrs Klarreich: I am having alot of trouble with related rates and optimization problems. Is there anyway you could provide small guide on how to approach these problems ?

Thank you for your time

Answer
---------- Your question: -----------------
Dear Mrs Klarreich: I am having alot of trouble with related rates and optimization problems. Is there anyway you could provide small guide on how to approach these problems ?

Thank you for your time

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I am terribly sorry, but Mrs Klarreich is unavailable to answer questions at this moment.  Nor are my children, Ms E. Klarreich or Ms N. Klarreich, both of whom are busy with young infants.

So you are stuck with me.  I'll do the best I can.

About related rate problems:  Would you believe you are the first person to ask about them?

No?

Well, would you believe you are the 19th person to ask about them?

Probably.  They aren't easy.  If you are having trouble, there is nothing wrong with you.  So you have to stick with them just like everyone else.  This is applied math, and if you are studying some science or engineering or other applied subject, THIS IS WHAT YOU ARE HERE TO LEARN.
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In any related rate problem, you will want to set up a framework like this:

Part I. Choose variables.  Decide what quantities in the problem are changing, and give them names.  Remember that your algebra teacher told you not to get lazy in this part.  For example, when you had a problem like "Three years ago Mary was twice as old as Ann." you should not write:

    Let x = Mary

but instead write something like:

    Let M = Mary's age three years ago.

So you will have two OR MORE variables, being very exact and specific about their meanings.

Part II. Express the rates of change of these variables as derivatives with respect to time.  (Time is NOT one of the variables you write down, but it is there nonetheless.)

For example, if one of the variables is a volume, you will write:

dV/dt = the rate at which:
          sand is falling onto the pile, or
          water is flowing into the tank, or
          air is escaping from the balloon, etc....

Indicate which of these rates are given in the problem, and which one is to be found.

Part III. Find some relationship among these variables.  For this, you just have to use your life experience, look up formulas, draw diagrams (I mean really, really, careful diagrams) etc.

Part IV. Differentiate that relation.  Using implicit differentiation with respect to time, your rates that you wrote in Part I pop up in this step.

Part V. NOW substitute any numerical facts given.  If the problem says "Find the rate..... when the radius is 12," here is where it gets used.  Do whatever algebra

is needed to find the rate you were asked for.

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In 'optimization' problems (commonly known as Max-min) you will also have variables to choose.  Here, however, there usually isn't any 'time' involved, just those variables.  Typically, you want to:

Choose the value of one variable, so that

some other variable is 'optimized'.

So you express the variable to be optimized in terms of the other variable.  Sometimes there is more than one other variable.  Then you need some fact that lets you eliminate all but one. ("the radius is always half of the height")  So finally, you write something like:

V = some function of (r)

meaning you want to find a value of 'r' that maximizes (minimizes) V.

Once you get this far, the rest is downhill.  (maybe a bumpy ride downhill, but you've done the hard part)

You differentiate, to get  dV/dr.
You set the derivative = 0.
You solve for r.

You take your value(s) of r (maybe more than one) and substitute back into the function to get your optimum value.  You ALSO check back with any endpoints for the practical situation.  One of those is the answer.