Chap 16 Exercises

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Class activity

Exercise 1

Figure 1: Power produced by the author’s rooftop photovoltaic array in June 4 (solid line) and June 3 (bars), 2022.

Figure fig-solar-june-c2 comes from the monitoring app for a photovoltaic (solar electricity) array on two consecutive days in June 2022. These data were presented first in Exercise exr-KINWgK, where you were asked to model the June 3 pattern as a bump function solar(t) multiplied by an irregular function cloudiness(t) with a domain between 0 (full cloudiness) and 1 (no clouds). The June 4 pattern is taken as the bump solar(t).

With that multiplicative model in mind, take a closer look at Figure fig-solar-june-c2, particularly for the early part of the day. In the interval from 6 to 9 am, the June 4 power production is below that of June 3.

How is this inconsistent with the “power(t) \(\equiv\) solar(t) \(\times\) cloudiness(t)” model?
The June 4 data starts low and then increases steeply around 9am. What evidence is there that the actual solar(t) function has a smoother increase?
Having in mind the modeling cycle, how might you modify the model simply to make it consistent with the data? (Hint: Is June 4 really a bump?)

Note: At the end of the day, both the June 3 and June 4 data show an extra bump of a sort not seen in the pattern-book gaussian bump function. Potentially this could be due to cloudiness, but the amount of cloudiness would have to be exactly the same for June 3 and 4, which seem unlikely. More realistically, the bump represents an earlier fall off due to shadows of trees as the evening sun get low in the sky.

Exercise 2 Watch the movie in Figure fig-kitten-find-linen.

Bacteria exhibit exponential growth under optimal conditions. In general, if the rate of growth depends on some quantity (here bacteria) then the exponential is the best first guess at a model. In the movie, notice that the rate of expansion depends on the number of bacteria present; the more E.coli, the faster the rate of growth. This is true for any exponential process: the instantaneous rate of growth or decay depends on the amount currently present.

If the experiment were continued indefinitely, the number of bacteria would eventually outgrow the petri dish or deplete their food source. When this happens, we say the bacteria have approached the carrying capacity of their environment. When the population is constrained in this way, a sigmoid would be a more appropriate model to start your modeling process. So, the deciding factor between exponential and sigmoid really depends upon whether 1) we assume a constrained or unconstrained environment, and 2) we let the bacteria reach the carrying capacity of the petri dish or not.

Is there any obvious sign that the bacteria are reaching the carrying capacity of their environment before the end of the movie?

question id: kitten-find-linen-1

Estimate the doubling time of the number of bacteria as they are growing exponentially. Do this by figuring out how long it takes the area of the colony to double (roughly). Hint: You’ll need to use the time marker in the bottom left corner of the movie.

question id: kitten-find-linen-2

Estimate the doubling time in another way, by observing an individual bacterium. At any point in the movie, choose a bacterium at random. Watch it until it splits in two then, immediately, note the time. Watch some more until one of the two children split. The time difference between the mother’s split and the child’s split is the doubling time.

question id: kitten-find-linen-3

Compare the doubling time for a mother in the center of a large colony to the doubling time of a mother on the edge of the colony. Is there any clear sign that growth in the more crowded part of the colony is slower than in the suburbs?

question id: kitten-find-linen-4

Credit: Math 141Z/142Z 2021-2022 development team.

Exercise 3 A person breathes in and out roughly every three seconds. The volume \(V\) of air in the person’s lungs varies between a minimum of \(2\) liters and a maximum of \(4\) liters. Assume time \(t\) is measured in seconds.

Remember that a full cycle of the sine wave \(\sin(x)\) involves \(x\) going from its starting value to that value plus \(2 \pi\).

Which of the following is the most appropriate of these models for \(V(t)\)?

\(V(t) \equiv 2 \sin \left( \frac{\pi}{3} t \right) + 2\)

\(V(t) \equiv \sin \left( \frac{2\pi}{3} t \right) + 3\)

\(V(t) \equiv 2 \sin \left( \frac{2\pi}{3} t \right)+ 2\)

\(V(t) \equiv \sin \left( \frac{\pi}{3} t \right) + 3\)

question id: lamb-talk-gloves-QA1

A respiratory cycle can be divided into two parts: inspiration and expiration. Please do an experiment. Using a clock or watch, breath with a total period of 3 seconds/breath, that is, complete one breath every three seconds. Once you have practiced this and can do it without forcing either phase of breathing, make a rough estimate of what fraction of the cycle is inspiration and what fraction is expiration. (The “correct/incorrect” answers here are right for most people. Your natural respiration might be different.)

Which is true?

Inspiration lasts longer than expiration

Expiration lasts longer than inspiration

Inspiration and expiration each consume about the same fraction of the complete cycle.

question id: lamb-talk-gloves-QA2

Class activity

Mini modeling project