Jan 13

## Exponential Decay and Drug Interactions

Working together with my best friend since the 5th grade who is now a nurse practitioner, I developed this problem in which students have to use exponential decay (half life) to protect their patient from dangerous drug interactions.

Your patient has a fungal infection and is taking ketoconazole 400mg daily for 10 days. During this time they have been advised to stop taking their atorvastatin because of the potential for severe interaction. They may begin taking the atorvastatin again when the concentration of ketoconazole is at 12.5% of the original amount. The half life of ketoconazole is 8 hours.

How many hours after the last dose of ketoconazole can you patient resume taking the atorvastatin?

Jun 18

## Designing Toys

My son has a set of construction toys called K’nex.  In this problem I ask my Elementary Functions (Precalculus) students to think like the designers of these toys.  I have written two versions of this activity.  The file Knex-Pyth-Comp contains the pdf of both versions.  Contact me by email or on Twiter @ArcheyDawn if you would like the .tex file.  The file begins with a shared introduction.  Problem 1 (which can stand alone) is just one application of the Pythagorean Theorem plus a little thinking.  Problem 2 (which can stand alone) requires using the Pythagorean Theorem repeatedly, function composition, and (of course) thinking.

Introduction

The main pieces of the set are sticks of various lengths and connectors that look like gears to hook the sticks together. To make interesting shapes, several different sizes of sticks will be needed.  My son’s set of K’nex has 5 different lengths, pictured below.  From smallest to largest, they are black, yellow, gray, red, and purple.  When I do this problem with my class, I will bring the K’nex with me so that the students can measure and see if they are getting the correct answers.

One of the basic shapes that will be used by children to construct a variety of objects is an isosceles right triangle.  You can see an example of a bicycle built from K’nex in the picture below.  Notice the two isosceles right triangles making the frame of the bike.

Problem 1 (just uses Pythagorean Theorem)

If the length of the gray bars which will form the legs of the isosceles right triangle is 4.1 cm and if it is 0.6 cm from the end of the bar to the center of the connector (the vertex of the triangle), how long should the red bars be made which will form the hypotenuse of the triangle.  See the picture below for reference.

Problem 2 (uses Pythagorean theorem and function composition)

Let x be the length of the legs of the smallest isosceles right triangle it will be possible to make with your construction toy.  These legs will consist of a black stick with a connector on each end.  The distance from the end of the stick to the center of the connector (the vertex of the triangle) is 0.6 cm.  The hypotenuse of this smallest triangle will consist of a yellow stick attached to the connectors.  See the picture below.  Give all answers in a complete sentence with units.

a)  (First lets try to design the set of toy’s the easy way.) The easiest thing to do would be to pick integer lengths for the lengths of the sticks.  Suppose the set of toys is manufactured so that the black sticks has length 2 and the yellow sticks have length 3.  Can an isosceles right triangle be built from such a set of toys?  Why or why not?

b) (Ok, so the easy way didn’t work.  For the rest of the worksheet, lets try being more careful.) What is the length of the black stick?  (Give a formula in terms of x).  For the rest of this problem, it will be important to keep track of whether we are discussing the length of a stick or the length of the whole side of the triangle.  You might find it useful to record your answers in the table included at the end of the worksheet so that you can refer back to them as you work.

c)  Give a function f(x) for the length of the hypotenuse of a triangle whose legs have length x.

d) Suppose you decide to make the black sticks 1.4 cm long.  What is the length of the legs of the isosceles right triangle that can be made with them as shown in the picture?  What is the length of the hypotenuse of that triangle?

e) Still supposing you decide to make the black sticks 1.4 cm long, what is the length of the hypotenuse of the triangle described in the previous problem? Compute the length of the hypotenuse in two ways and make sure you get the same answer both times.  %using f(x) and usin the Pythagorean theorem directly.

f)  Go back to assuming the length of the legs (built from black sticks) of the smallest triangle is the unknown x.  Recall that the length of the hypotenuse (built from a yellow stick) of this triangle is given by f(x).  Suppose you wanted to allow bigger isosceles right triangles to be built.  If the legs are built from yellow sticks, use the Pythagorean theorem to give a formula, in terms of x, for the length of the hypotenuse of the triangle with legs made from yellow sticks.  We will use gray sticks for this larger hypotenuse.  The following picture shows the whole set of possible isosceles right triangles.

g)  Compute $f \circ f (x)$ and compare your answer to the formula in the previous problem.  (You may need to do some simplification in order to compare well.)  Explain why any similarities you noticed make sense.

h) Suppose you want to make a third size triangle, this one with legs formed from the gray sticks and a hypotenuse which we will make red.  Does $f \circ f \circ f (x)$ have any bearing on this problem, if so what?

i) Go back to assuming the black sticks will be 1.4 cm long?  How long will the gray sticks be?  How long will the red sticks be?

j)  Still assuming the black sticks are 1.4 cm long, how long will the hypotenuse be of the triangle whose legs are made from red sticks?  How long should we make the purple sticks we will use to form this hypotenuse?

Jul 27

## Pluto

I was so excited about New Horizon’s fly by of Pluto, that I decided to create some math problems about it.  Here they are.  The first one uses the concepts of percent change and volume of a sphere.  The second two use right triangle trigonometry.

1) Answer the question posed in @MathInTheNews ‘s tweet from July 14, 2015:
New Horizons says Pluto’s diameter is 1473mi, 50mi larger than believed. What % does that change its volume estimate?

(Yes, the picture was part of the tweet)

2) Pick two mountains in the large picture of Pluto.  The angle of elevation from the tip of the mountain’s shadow to the tip of the mountain is approximately y/736.5 radians where y is the distance in miles from the base of the mountain to the edge of Pluto in the picture.

a)  What is the height of each of your two mountains?

b) How precise do you think our method is?

c)  Do you think NASA’s claim that there are 11,000 ft high mountains on Pluto is credible?

(Note to other instructors, the estimation of the angle given above is very imprecise, so the students ended up finding mountain heights no more than 7,000 feet high.  In order to improve precision, I printed the following picture out as a 11 by 17 picture).

3) In a July 13, 2015 article for Reuter’s Irene Klotz wrote, “Mysterious Pluto looms large and turns out to be larger than expected as NASA’s New Horizons spacecraft wraps up a nearly decade-long journey, with a close flyby on track for Tuesday, scientists said on Monday.  The nuclear-powered probe was in position to pass dead center of a 60-by-90-mile (97-by-145 km) target zone between the orbits of Pluto and its primary moon, Charon, at 7:49 a.m. EDT (1149 GMT) on Tuesday, said managers at New Horizons mission control center, located at the Johns Hopkins University Applied Physics Laboratory outside of Baltimore. After a journey of 3 billion miles (4.88 billion km), threading that needle is like golfer in New York hitting a hole-in-one in Los Angeles, project manager Glen Fountain told reporters.”

Is this analogy correct?  (You will need to look up some additional numbers and make additional assumptions to answer this question).

Jul 07

## Global Temperature-meaning of Slope and y-intercept

This is a problem I work through in my pre-calculus class when we are discussing the meaning of the slope and y-intercept.  It also gives me the opportunity to discuss the perils of extrapolating far outside of the data set.

Consider the following data and the linear model that fits it.

The equation for the linear model is y=0.019x+4.76.  Answer the following questions in complete sentences with units.

1. What are the units for the slope?
2. What does the slope mean in this particular situation?
3. What are the units for the y-intercept?
4. What does the y-intercept mean in this particular situation?

(data from http://www.esrl.noaa.gov/gmd/dv/data/index.php?site=mlo&parameter_name=Carbon%2BDioxide&frequency=Monthly%2BAverages and http://www.ncdc.noaa.gov/cag/ )

Mar 18

## Roots, factorials, trig, and summation notation

Posted in Comic Strips,

In this FoxTrot strip Paige has been cramming for her math final and thus is answering all questions in mathematical notation.

http://assets.amuniversal.com/b07d90305e30012ee3bf00163e41dd5b

(Its the May 22, 2005 strip, if the link doesn’t work.)

In precalculus I asked the students:

1) What temperature will it be tomorrow?  Is it going to be hot tomorrow?

(The answer is sin^{-1}(1) =90 degrees, so it will be hot).

2) What does the girl (Paige) want for snack?

(The answer is 3.14159265359…, so she wants pie).

In calculus of sequences and series, you could also ask what the discount was in the fourth panel, which requires computing the sum of an infinite geometric series.

In an algebra class one could ask what time Paige went to bed (\sqrt{121}= 11:00) or What is on TV? (4!=24).

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