### Subsection3.3.4Exercises

#### SubsubsectionExercises for § 3.3.1

###### 1

Which of the following is a differential equation for an unknown function $y$ of $x\text{?}$

\begin{align*} \amp\mbox{(a) } y=\diff{y}{x} \amp \amp\mbox{(b) } \diff{y}{x}=3\left[y-5\right] \amp \amp\mbox{(c) } y=3\left[y-\diff{x}{x}\right]\\ \amp\mbox{(d) } e^x=e^y+1 \amp \amp\mbox{(e) } y=10e^x \end{align*}
###### 2

Which of the following functions $Q(t)$ satisfy the differential equation $Q(t)=5\ds\diff{Q}{t}\text{?}$

\begin{align*} \amp\mbox{(a) } Q(t)=0\amp \amp\mbox{(b) } Q(t)=5e^t\amp \amp\mbox{(c) } Q(t)=e^{5t}\\ \amp\mbox{(d) } Q(t)=e^{t/5}\amp \amp\mbox{(e) } Q(t)=e^{t/5}+1 \end{align*}
###### 3

Suppose a sample starts out with $C$ grams of a radioactive isotope, and the amount of the radioactive isotope left in the sample at time $t$ is given by

\begin{equation*} Q(t)=Ce^{-kt} \end{equation*}

for some positive constant $k\text{.}$ When will $Q(t)=0\text{?}$

###### 4(✳)

Consider a function of the form $f(x) = A e^{kx}$ where $A$ and $k$ are constants. If $f(0)=5$ and $f(7)=\pi\text{,}$ find the constants $A$ and $k\text{.}$

###### 5(✳)

Find the function $y(t)$ if $\ds\diff{y}{t} +3y = 0\text{,}$ $y(1) = 2\text{.}$

###### 6

A sample of bone belongs to an animal that died 10,000 years ago. If the bone contained 5 $\mu$g of Carbon-14 when the animal died, how much Carbon-14 do you expect it to have now?

###### 7

A sample containing one gram of Radium-226 was stored in a lab 100 years ago; now the sample only contains 0.9576 grams of Radium-226. What is the half-life of Radium-226?

###### 8(✳)

The mass of a sample of Polonium--210, initially 6 grams, decreases at a rate proportional to the mass. After one year, 1 gram remains. What is the half--life (the time it takes for the sample to decay to half its original mass)?

###### 9

Radium-221 has a half-life of 30 seconds. How long does it take for only 0.01% of an original sample to be left?

###### 10

Polonium-210 has a half life of 138 days. What percentage of a sample of Polonium-210 decays in a day?

###### 11

A sample of ore is found to contain $7.2 \pm 0.3\;\mu$g of Uranium-232, the half-life of which is between 68.8 and 70 years. How much Uranium-232 will remain undecayed in the sample in 10 years?

#### SubsubsectionExercises for § 3.3.2

###### 1

Which of the following functions $T(t)$ satisfy the differential equation $\ds\diff{T}{t}=5\left[T-20\right]\text{?}$

\begin{align*} \amp\mbox{(a) } T(t)=20 \amp \amp\mbox{(b) } T(t)=20e^{5t}-20 \amp \mbox{(c) } T(t)=e^{5t}+20\\ \amp\mbox{(d) } T(t)=20e^{5t}+20 \end{align*}
###### 2

At time $t=0\text{,}$ an object is placed in a room, of temperature $A\text{.}$ After $t$ seconds, Newton's Law of Cooling gives the temperature of the object is as

\begin{equation*} T(t)=35e^{Kt}-10 \end{equation*}

What is the temperature of the room? Is the room warmer or colder than the object?

###### 3

A warm object is placed in a cold room. The temperature of the object, over time, approaches the temperature of the room it is in. The temperature of the object at time $t$ is given by

\begin{equation*} T(t)=[T(0)-A]e^{Kt}+A. \end{equation*}

Can $K$ be a positive number? Can $K$ be a negative number? Can $K$ be zero?

###### 4

Suppose an object obeys Newton's Law of Cooling, and its temperature is given by

\begin{equation*} T(t)=[T(0)-A]e^{kt}+A \end{equation*}

for some constant $k\text{.}$ At what time is $T(t)=A\text{?}$

###### 5

A piece of copper at room temperature (25$^\circ$) is placed in a boiling pot of water. After 10 seconds, it has heated to 90$^\circ\text{.}$ When will it be 99.9$^\circ\text{?}$

###### 6

Today is a chilly day. We heated up a stone to 500$^\circ$ C in a bonfire, then took it out and left it outside, where the temperature is 0$^\circ$ C. After 10 minutes outside of the bonfire, the stone had cooled to a still-untouchable 100$^\circ$ C. Now the stone is at a cozy 50$^\circ$ C. How long ago was the stone taken out of the fire?

###### 7(✳)

Isaac Newton drinks his coffee with cream. To be exact, 9 parts coffee to 1 part cream. His landlady pours him a cup of coffee at $95^\circ$ C into which Newton stirs cream taken from the icebox at $5^\circ$ C. When he drinks the mixture ten minutes later, he notes that it has cooled to $54^\circ$ C. Newton wonders if his coffee would be hotter (and by how much) if he waited until just before drinking it to add the cream. Analyze this question, assuming that:

1. The temperature of the dining room is constant at $22^\circ$ C.
2. When a volume $V_1$ of liquid at temperature $T_1$ is mixed with a volume $V_2$ at temperature $T_2\text{,}$ the temperature of the mixture is $\dfrac{V_1T_1+V_2T_2}{V_1+V_2}\text{.}$
3. Newton's Law of Cooling: The temperature of an object cools at a rate proportional to the difference in temperature between the object and its surroundings.
4. The constant of proportionality is the same for the cup of coffee with cream as for the cup of pure coffee.
###### 8(✳)

The temperature of a glass of iced tea is initially $5^\circ\text{.}$ After 5 minutes, the tea has heated to $10^\circ$ in a room where the air temperature is $30^\circ\text{.}$

1. Use Newton's law of cooling to obtain a differential equation for the temperature $T(t)$ at time $t\text{.}$
2. Determine when the tea will reach a temperature of $20^\circ\text{.}$
###### 9

Suppose an object is changing temperature according to Newton's Law of Cooling, and its temperature at time $t$ is given by

\begin{equation*} T(t)=0.8^{kt}+15 \end{equation*}

Is $k$ positive or negative?

#### SubsubsectionExercises for § 3.3.3

###### 1

Let a population at time $t$ be given by the Malthusian model,

\begin{equation*} P(t)=P(0)e^{bt}\mbox{ for some positive constant } b. \end{equation*}

Evaluate $\ds\lim_{t \to \infty}P(t)\text{.}$ Does this model make sense for large values of $t\text{?}$

###### 2

In the 1950s, pure-bred wood bison were thought to be extinct. However, a small population was found in Canada. For decades, a captive breeding program has been working to increase their numbers, and from time to time wood bison are released to the wild. Suppose in 2015, a released herd numbered 121 animals, and a year later, there were 136  14 These numbers are loosely based on animals actually released near Shageluk, Alaska in 2015. Watch the first batch being released here.. If the wood bison adhere to the Malthusian model (a big assumption!), and if there are no more releases of captive animals, how many animals will the herd have in 2020?

###### 3

A founding colony of 1,000 bacteria is placed in a petri dish of yummy bacteria food. After an hour, the population has doubled. Assuming the Malthusian model, how long will it take for the colony to triple its original population?

###### 4

A single pair of rats comes to an island after a shipwreck. They multiply according to the Malthusian model. In 1928, there were 1,000 rats on the island, and the next year there were 1500. When was the shipwreck?

###### 5

A farmer wants to farm cochineals, which are insects used to make red dye. The farmer raises a small number of cochineals as a test. In three months, a test population of cochineals will increase from 200 individuals to 1000, given ample space and food.

The farmer's plan is to start with an initial population of $P(0)$ cochineals, and after a year have $1\,000\,000+P(0)$ cochineals, so that one million can be harvested, and $P(0)$ saved to start breeding again. What initial population $P(0)$ does the Malthusian model suggest?

###### 6

Let $f(t)=100e^{kt}\text{,}$ for some constant $k\text{.}$

1. If $f(t)$ is the amount of a decaying radioactive isotope in a sample at time $t\text{,}$ what is the amount of the isotope in the sample when $t=0\text{?}$ What is the sign of $k\text{?}$
2. If $f(t)$ is the number of individuals in a population that is growing according to the Malthusian model, how many individuals are there when $t=0\text{?}$ What is the sign of $k\text{?}$
3. If $f(t)$ is the temperature of an object at time $t\text{,}$ given by Newton's Law of Cooling, what is the ambient temperature surrounding the object? What is the sign of $k\text{?}$

#### SubsubsectionFurther problems for § 3.3

###### 1(✳)

Find $f(2)$ if $f'(x) = \pi f(x)$ for all $x\text{,}$ and $f(0) = 2\text{.}$

###### 2

Which functions $T(t)$ satisfy the differential equation $\ds\diff{T}{t}=7T+9\text{?}$

###### 3(✳)

It takes 8 days for 20% of a particular radioactive material to decay. How long does it take for 100 grams of the material to decay to 40 grams?

###### 4

A glass of boiling water is left in a room. After 15 minutes, it has cooled to 85$^\circ$ C, and after 30 minutes it is 73$^\circ$ C. What temperature is the room?

###### 5(✳)

A 25-year-old graduate of UBC is given $50,000 which is invested at 5% per year compounded continuously. The graduate also intends to deposit money continuously at the rate of$2000 per year. Assuming that the interest rate remains 5%, the amount $A(t)$ of money at time $t$ satisfies the equation

\begin{equation*} \diff{A}{t}= 0.05 A+2000 \end{equation*}
1. Solve this equation and determine the amount of money in the account when the graduate is 65.
2. At age 65, the graduate will withdraw money continuously at the rate of $W$ dollars per year. If the money must last until the person is 85, what is the largest possible value of $W\text{?}$
###### 6(✳)

An investor puts $120,000 which into a bank account which pays 6% annual interest, compounded continuously. She plans to withdraw money continuously from the account at the rate of$9000 per year. If $A(t)$ is the amount of money at time $t\text{,}$ then

\begin{equation*} \diff{A}{t}= 0.06 A-9000 \end{equation*}
1. Solve this equation for $A(t)\text{.}$
2. When will the money run out?
###### 7(✳)

A particular bacterial culture grows at a rate proportional to the number of bacteria present. If the size of the culture triples every nine hours, how long does it take the culture to double?

###### 8(✳)

An object falls under gravity near the surface of the earth and its motion is impeded by air resistance proportional to its speed. Its velocity $v$ satisfies the differential equation

\begin{equation*} \dfrac{dv}{dt}=-g-kv \end{equation*}

where $g$ and $k$ are positive constants.

1. Find the velocity of the object as a function of time $t\text{,}$ given that it was $v_0$ at $t=0\text{.}$
2. Find $\lim\limits_{t\rightarrow\infty} v(t)\text{.}$