Math – Applications for Living IX

In our Math119 course, we are studying models — linear (repeated adding) and exponential (repeated multiplying).  Although some of the details we are including are not very practical, some are practical … and helpful in understanding everyday numbers like ‘inflation’.

Here is a situation we looked at:

If prices increase at a monthly rate of 1.5%, by what percentage do they increase in a year?

Much of our work in class has been on translating from a “percent change” statement to a “multiplying statement”.  Most students saw that this 1.5% increase meant that the multiplier was 1.015.  To answer this question, we just evaluated

We did have a little struggle about using the resulting value (1.1956 …); with a little nudging, we agreed that the annual increase was 19.6%.  Even though we have done quite a few finance applications, this result was a little surprising … students thought we would multiply 0.015 by 12 (18%).

While we were working on models, we also introduced using a calculator procedure to find answers to ‘difficult’ questions [meaning that we used a numeric approach to solving exponential equations].  Take a look at this problem:

Fifty mg of a drug are administered at 2pm, and 20% of the drug is eliminated each hour.  When will it reach 10 mg in the body (the minimum effective level)?

We’ve got that percent change going on; students are generally getting that — this is a multiplier of 0.80.  [This problem is much tougher when I give them drug levels for consecutive 1 hour intervals … like after 3 hours and after 4 hours.].  We set up this equation

To solve this problem, we used a graphing calculator ‘intersect’ process … placing this function on ‘y1’ and the output we needed (10) on ‘y2’.   Our solution (about 7.2 hours)  is useful in understanding the frequency for some prescriptions (3 times per day in this case).  In class, we also approach this same problem as a ‘half-life’ situation; conceptually, that is more complex … and specialized, so we do not emphasize the half-life method.  [Half-life is mostly there to help students if they take a science course which uses half-life concepts.]

We also point out that the intersect process used here is very flexible; it may be one of the most practical things they get out of the course.

 
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