Modeling

Unit 1.2 Modeling

Mathematical modeling means writing down mathematics corresponding to a given real-world (physical, chemical, biological) scenario, along with equations or other relations that could be expected to hold, at least approximately or under further assumptions.

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Unpacking this, we see a number of features. First of all one must define mathematical objects in the model: variables, sets, functions, equations and so forth. Secondly, one must give interpretations of everything in the model. An interpretation tells what physical quantity is associated with each of the constants and variables and what relation is meant by each function. Real-world quantities include units, so this part always involves stating units.

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Thirdly, often one needs to add assumptions about the scenario. These say the circumstances under which would you expect the mathematics to be correct for the model. These assumptions are determined by the real world; they are not mathematical.

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Lastly, if there are questions given in the scenario, it is necessary to say what part of the mathematics answers the question(s). After this, what is left is a math problem: solve for the quantities that answer the questions.

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In the following example, we have underlined parts of the modeling exercise that reflect the outline we have given, such as naming of variables, interpretation, units and hypotheses.

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Example 1.7.

Scenario 🔗: Galileo observes that objects falling a short time seem to fall a distance that was proportional to the square of the time and independent of the object: 4 feet for an object falling half a second, 9 feet after three quarters of a second, 16 feet after one second, and so forth. Galileo decides to measure the Tower of Pisa by dropping a stone from the top of the tower and measuring the time it takes for him to hear it hit the ground. Make a model for this and use it to estimate the elapsed time Galileo measure between dropping the stone and hearing the sound.
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Model 🔗: Let \(f(t)\) be the distance in feet that an object falls in \(t \) seconds, starting from rest. The wording of the scenario tells us that \(f(t) = c \, t^2\) where \(c \) has units of feet per seconds squared. This assumes we set \(t=0 \) at the time of release and measure distance from the point of release. We are asked to determine \(t \) such that \(f(t) = h\text{,}\) where \(h \) is the height of the Tower of Pisa. Equivalently, we need to find \(f^{-1} (h)\text{.}\) We assume that the model is accurate. What that means in this case is that we can ignore things such as air resistance and the time lag for the sound of impact to get back to Galileo’s ear.
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Solution: 🔗: We look up the height of the Tower of Pisa to find that \(h = 186 \) feet. We solve for \(c \) given Galileo’s data for small distances and find that \(c = 16 \) (for example: \(f(1/2) = c (1/2)^2 = 4 \) implies \(c=16\)). We can solve directly for \(16 t^2 = 186 \) or we can compute the inverse function to \(f \) yielding \(f^{-1} (x) = \sqrt{x/16} \) and substitute 186 for \(x \text{.}\) Either way we get \(t = \sqrt{186/16} \approx 3.40 \text{.}\) It may sound pedantic, but probably we should justify our choice of the positive square root by saying the whole experiment only covers time after the release, that is, \(t \geq 0 \text{.}\)
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Were the physical assumptions warranted? Many objects would be slowed by air resistance over such a distance. Probably Galileo would have had to drop something like a rock in order for the fall not to have been significantly slowed. Looking up the speed of sound, it would take an extra \(1/4 \) second to register the sound. Probably Galileo could measure time to within greater accuracy that \(1/4 \) second, so this assumption is definitely shaky.

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Checkpoint 24.

Example 1.7 states two physical assumptions -- that air resistance and time lag of sound are both negligible. But there are more physical assumptions built into the model than just those. Identify as many as you can.

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For each physical assumption you identify, state whether it is reasonable or not.

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Using units to interpret a model.

Sometimes we’re given (or derive) a formula whose meaning isn’t immediately clear; often the units can be our guide here. As an example, consider the body mass index (BMI), which medical researchers and doctors frequently use as a measure of obesity. BMI was invented by Adolphe Quetelet, a Belgian statistician, in the mid-1800s; it’s still used today. (Think of how few medical tools from that period are still in use. . .)

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A patient’s BMI is computed by the following formula:

\begin{equation*} \text{BMI }=\frac{\text{mass}}{(\text{height})^2} \end{equation*}

If we measure mass in kilograms and height in meters, the units of BMI are \(kg/m^2\text{.}\) That doesn’t seem very revelatory.

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But now consider: for objects of uniform density (which humans aren’t, but close enough), we have

\begin{equation*} \text{mass } = \text{ density}\times\text{ volume} \end{equation*}

Substituting, we get

\begin{equation*} \text{BMI }=\text{ density}\times\frac{\text{volume}}{(\text{height})^2} \end{equation*}

Now, broadly speaking, most people have approximately the same density. So when we compare two patients’ BMIs, we’re really comparing the quantity

\begin{equation*} \frac{\text{volume}}{(\text{height})^2} \end{equation*}

which has units \(m^3/m^2=m\text{.}\) That is, roughly speaking BMI measures a length! But the length of what?

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Here’s were we need another rough biological fact: most peoples’ width is proportional to their height; that is,

\begin{equation*} \text{width }=k\times\text{ height} \end{equation*}

for some constant of proportionality \(k\text{.}\) If we pretend that a person’s shape is a rectangular prism, then the volume of this prism would be

\begin{equation*} \text{volume }=\text{ width }\times\text{ height }\times \text{ depth} \end{equation*}

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Because width is proportional to height, that works out to approximately

\begin{equation*} \text{volume }=k(\text{height})^2\times\text{ depth} \end{equation*}

so that the quantity \(\frac{\text{volume}}{(\text{height})^2}\) is really

\begin{equation*} \frac{\text{volume}}{(\text{height})^2}=\frac{k\times (\text{height})^2\times\text{ depth}}{(\text{height})^2}=k\times \text{ depth} \end{equation*}

That is to say, what BMI really measures is how deep -- perhaps better to say, how thick -- a person is.

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Figure 1.8. BMI assumes the body is basically a rectangular prism.
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Checkpoint 25.

In the discussion above, we made several biological assumptions in order to interpret BMI as a measure of “body thickness”. Identify each such assumption and critique it.

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Then think about the effect your critique would have on the usefulness of BMI as a measure of obesity.

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By way of example: one assumption is that all humans have approximately the same density. Bodybuilders tend to have more muscle than the average person, and muscle is denser than other soft tissues; so the density of a bodybuilder will be above average. Therefore, bodybuilders will have a higher BMI -- that is, they will appear to be more obese than they are when measured with BMI.

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