91Ó°ÊÓ

An object of mass \(m\) is thrown vertically upward with an initial velocity of \(u_{0}\). Air resistance is proportional to its instantaneous velocity with constant of proportionality \(k\). Show that the maximum height attained by the object is $$ \frac{m v_{0}}{k}-\frac{m^{2} g}{k^{2}} \ln \left(1+\frac{k v_{0}}{m g}\right) $$

Chemical Reactions In a certain chemical reaction a substance is converted into another substance at a rate proportional to the square of the amount of the first substance present at any time \(t\). Initially \((t=0), 50 \mathrm{~g}\) of the first substance was present; 1 hr later, only \(10 \mathrm{~g}\) of it remained. Find an expression that gives the amount of the first substance present at any time \(t\). What is the amount present after \(2 \mathrm{hr}\) ?

Carbon-14 Dating Skeletal remains of the so-called Pittsburgh Man unearthed in Pennsylvania had lost \(82 \%\) of the carbon14 they originally contained. Determine the approximate age of the bones. (The half-life of carbon C-14 is 5730 years.)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(y_{1}\) is a solution of the homogeneous equation \(y^{\prime}+P y=0\) associated with the nonhomogeneous equation \(y^{\prime}+P y=f\) and \(y_{2}\) is a solution of the nonhomogeneous equation, then \(y=c y_{1}+y_{2}\) is a solution of the nonhomogeneous equation, where \(c\) is any constant.

Effect of Immigration on Population Growth Suppose that a country's population at any time \(t\) grows in accordance with the rule $$ \frac{d P}{d t}=k P+I $$ where \(P\) denotes the population at any time \(t, k\) is a positive constant reflecting the natural growth rate of the population, and \(I\) is a constant giving the (constant) rate of immigration into the country. a. If the total population of the country at time \(t=0\) is \(P_{0}\), find an expression for the population at any time \(t\). b. The population of the United States in the year 1980 \((t=0)\) was \(226.5\) million. Suppose that the natural growth rate is \(0.8 \%\) annually \((k=0.008)\) and that net immigration is allowed at the rate of \(0.5\) million people per year \((I=0.5) .\) What will the U.S. population be in \(2010 ?\)

A Falling Raindrop As a raindrop falls, it picks up more moisture, and as a result, its mass increases. Suppose that the rate of change of its mass is directly proportional to its current mass. a. Using Newton's Law of Motion, \(\frac{d}{d t}(m v)=F=m g\), where \(m(t)\) is the mass of the raindrop at time \(t, v\) is its velocity (positive direction is downward), and \(g\) is the acceleration due to gravity, derive the (differential) equation of motion of the raindrop. b. Solve the differential equation of part (a) to find the velocity of the raindrop at time \(t\). Assume that \(v(0)=0\). c. Find the terminal velocity of the raindrop, that is, find \(\lim _{t \rightarrow \infty} v(t)\)

Discharging Water from a Tank A container that has a constant cross section \(A\) is filled with water to height \(H\). The water is discharged through an opening of cross section \(B\) at the base of the container. By using Torricelli's Law, it can be shown that the height \(h\) of the water at time \(t\) satisfies the initialvalue problem $$ \frac{d h}{d t}=-\frac{B}{A} \sqrt{2 g h} \quad h(0)=H $$ a. Find an expression for \(h\). b. Find the time \(T\) it takes for the tank to empty. c. Find \(T\) if \(A=4\left(\mathrm{ft}^{2}\right), B=1\left(\mathrm{in} .^{2}\right), H=16(\mathrm{ft})\), and \(g=32\left(\mathrm{ft} / \mathrm{sec}^{2}\right)\)

Doomsday Equation Suppose that the population \(P\) satisfies the differential equation \(d P / d t=k P^{1.0 i}\), where \(k\) is a positive constant and \(P(0)=1\). a. Solve the initial-value problem. b. Suppose that \(k=0.1\). Plot the graph of \(P(t)\). c. Why is \(d P / d t=k P^{1.01}\) called the "doomsday equation"?

Spread of Disease A simple mathematical model in epidemiology for the spread of a disease assumes that the rate at which the disease spreads is jointly proportional to the number of infected people and the number of uninfected people. Suppose that there are a total of \(N\) people in the population, of whom \(N_{0}\) are infected initially. Show that the number of infected people after \(t\) weeks, \(x(t)\), is given by $$ x(t)=\frac{N}{1+\left(\frac{N-N_{0}}{N_{0}}\right) e^{-k N t}} $$ where \(k\) is a positive constant.

Von Bertalanffy Growth Model The von Bertalanffy growth model is used to predict the length of commercial fish. The model is described by the differential equation $$ \frac{d x}{d t}=k(L-x) $$ where \(x(t)\) is the length of the fish at time \(t, k\) is a positive constant called the von Bertalanffy growth rate, and \(L\) is the maximum length of the fish. a. Find \(x(t)\) given that the length of the fish at \(t=0\) is \(x_{0}\). b. At the time the larvae hatch, the North Sea haddock are about \(0.4 \mathrm{~cm}\) long, and the average haddock grows to a length of \(10 \mathrm{~cm}\) after 1 year. Find an expression for the length of the North Sea haddock at time \(t\). c. Plot the graph of \(x\). Take \(L=100(\mathrm{~cm})\). d. On average, the haddock that are caught today are between \(40 \mathrm{~cm}\) and \(60 \mathrm{~cm}\) long. What are the ages of the haddock that are caught?