ASSIGNMENT 2 Learning goals · Use mathematical models to formalize ideas about natural processes. · Proficiency with differential equations and exponential functions. . Practice reasoning with the results of your model analysis in the context of the phenomenon being modelled. · Practice graphing functions. Contributors On the first page of your submission, list the student numbers and full names (with the last name in bold) of all team members. Indicate members who have not contributed using the comment "(non-contributing)". Reflection question Reflection questions encourage you to think about how mathematics is done. This is an important ingredient of success. Reflection questions contribute to your engagement grade. 1. Consider the differences between the types of questions on the previous written assignment, Assignment 1, and the types of questions on your high school math assignments. In one or two paragraphs, describe one major difference, and explain how you might approach assignments in this course differently. Be specific and provide details. Assignment questions The questions in this section contribute to your assignment grade. Stars indicate the difficulty of the questions, as described on Canvas. In these questions, you will explore a simple model from the theory of pharmacokinetics which is what underlies the instructions on medications that tell you how much of the medication to take and how often. For example, the label on acetaminophen recommends something like 500 mg every 4 hours. To keep track of how much of a medication is in a person's blood, we derive a differential equation (DE) for the mass of the medication m in the blood at time t (see the derivation given below the assignment questions). 2. **** The rate constant for the clearance of acetaminophen (Tylenol) from the body of a typical human is r = 23 % per hour or equivalently r = 0.23 per hour. Suppose a person has no acetaminophen in their body at t = 0 when they take a 500 mg pill. (a) What is the DE for the mass m of acetaminophen remaining in the body at time t? (b) What is the initial condition (IC) for m(t)? (c) Solve the DE from (a) with the IC from (b) and determine how much acetaminophen remains in the body after 4 hours (the time recommended before taking another pill). (d) If the person takes another pill at t = 4 hours, we can define a new initial condition and DE whose solution will give the mass of acetaminophen remaining in the body at time t for t > 4 hours. What is the new initial condition: m2(4) =? What is the DE? Solve the DE with this IC. How much acetaminophen remains in the body at t = 8? 3. **** Suppose a person takes a pill with p mg of a medication in it every T hours, starting with the first pill at t = 0 and continuing indefinitely. The medication is cleared from the body with a rate constant of k measured in units of hr"1. Call