Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

Chapter 5

Applications of the Exponential and Natural Logarithm Functions - all with Video Answers

Educators

Section 1

Exponential Growth and Decay

Problem 1

Let $P(t)$ be the population (in millions) of a certain city $t$ years after 1990, and suppose that $P(t)$ satisfies the differential equation
$$
P^{\prime}(t)=.02 P(t), \quad P(0)=3
$$
(a) Find the formula for $P(t)$.
(b) What was the initial population, that is, the population in $1990 ?$
(c) What is the growth constant?
(d) What was the population in $1998 ?$
(e) Use the differential equation to determine how fast the population is growing when it reaches 4 million people.
(f) How large is the population when it is growing at the rate of 70,000 people per year?

Joseph Liao

Joseph Liao

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Problem 2

Approximately 10,000 bacteria are placed in a culture. Let $P(t)$ be the number of bacteria present in the culture after $t$ hours, and suppose that $P(t)$ satisfies the differential equation
$$
P^{\prime}(t)=.55 P(t)
$$
(a) What is $P(0)$ ?
(b) Find the formula for $P(t)$.
(c) How many bacteria are there after 5 hours?
(d) What is the growth constant?
(e) Use the differential equation to determine how fast the bacteria culture is growing when it reaches $100,000 .$
(f) What is the size of the bacteria culture when it is growing at a rate of 34,000 bacteria per hour?

Joseph Liao

Numerade Educator

Problem 3

After $t$ hours there are $P(t)$ cells present in a culture, where $P(t)=5000 e^{.2 t}$.
(a) How many cells were present initially?
(b) Give a differential equation satisfied by $P(t)$.
(c) When will the population double?
(d) When will 20,000 cells be present?

Nick Johnson

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Problem 4

The size of a certain insect population is given by $P(t)=300 e^{.01 t}$, where $t$ is measured in days.
(a) How many insects were present initially?
(b) Give a differential equation satisfied by $P(t)$.
(c) At what time will the population double?
(d) At what time will the population equal $1200 ?$

Joseph Liao

Numerade Educator

Problem 5

Determine the growth constant of a population that is growing at a rate proportional to its size, where the population doubles in size every 40 days.

Joseph Liao

Numerade Educator

Problem 6

Determine the growth constant of a population that is growing at a rate proportional to its size, where the population triples in size every 10 years.

Joseph Liao

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Problem 7

A population is growing exponentially with growth constant $.05 .$ In how many years will the current population triple?

Joseph Liao

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Problem 8

A population is growing exponentially with growth constant.04. In how many years will the current population double?

Joseph Liao

Numerade Educator

Problem 9

The rate of growth of a certain cell culture is proportional to its size. In 10 hours a population of 1 million cells grew to 9 million. How large will the cell culture be after 15 hours?

Joseph Liao

Numerade Educator

Problem 10

The world's population was $5.51$ billion on January 1,1993 , and $5.88$ billion on January 1,1998 . Assume that, at any time, the population grows at a rate proportional to the population at that time. In what year will the world's population reach 7 billion?

Joseph Liao

Numerade Educator

Problem 11

At the beginning of $1990,20.2$ million people lived in the metropolitan area of Mexico City, and the population was growing exponentially. The 1995 population was 23 million. (Part of the growth is due to immigration.) If this trend continues, how large will the population be in the year 2010 ?

Joseph Liao

Numerade Educator

Problem 12

The population (in millions) of a state $t$ years after 1970 is given by the graph of the exponential function $y=P(t)$ with growth constant. 025 in Fig. 6. [In parts (c) and (d) use the differential equation satisfied by $P(t) .]$
(a) What was the population in $1974 ?$
(b) When was the population 10 million?
(c) How fast was the population growing in $1974 ?$
(d) When is the population growing at the rate of 275,000 people per year?

Joseph Liao

Numerade Educator

Problem 13

A sample of 8 grams of radioactive material is placed in a vault. Let $P(t)$ be the amount remaining after $t$ years, and let $P(t)$ satisfy the differential equation $P^{\prime}(t)=-.021 P(t)$.
(a) Find the formula for $P(t)$.
(b) What is $P(0)$ ?
(c) What is the decay constant?
(d) How much of the material will remain after 10 years?
(e) Use the differential equation to determine how fast the sample is disintegrating when just 1 gram remains.
(f) What amount of radioactive material remains when it is disintegrating at the rate of $.105$ gram per year?
(g) The radioactive material has a half-life of 33 years. How much will remain after 33 years? 66 years? 99 years?

Joseph Liao

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Problem 14

226 is used in cancer radiotherapy, as a neutron source for some research purposes, and as a constituent of luminescent paints. Let $P(t)$ be the number of grams of radium 226 in a sample remaining after $t$ years, and let $P(t)$ satisfy the differential equation
$$
P^{\prime}(t)=-.00043 P(t), \quad P(0)=12 .
$$
(a) Find the formula for $P(t)$.
(b) What was the initial amount?
(c) What is the decay constant?
(d) Approximately how much of the radium will remain after 943 years?
(e) How fast is the sample disintegrating when just 1 gram remains? Use the differential equation.
(f) What is the weight of the sample when it is disintegrating at the rate of $.004$ gram per year?
(g) The radioactive material has a half-life of about 1612 years. How much will remain after 1612 years? 3224 years? 4836 years?

Joseph Liao

Numerade Educator

Problem 15

A person is given an injection of 300 milligrams of penicillin at time $t=0 .$ Let $f(t)$ be the amount (in milligrams) of penicillin present in the person's bloodstream $t$ hours after the injection. Then, the amount of penicillin decays exponentially, and a typical formula is $f(t)=300 e^{-.6 t}$.
(a) Give the differential equation satisfied by $f(t)$.
(b) How much will remain at time $t=5$ hours?
(c) What is the biological half-life of the penicillin (that is, the time required for half of a given amount to decompose) in this case?

Joseph Liao

Numerade Educator

Problem 16

Ten grams of a radioactive substance with decay constant $.04$ is stored in a vault. Assume that time is measured in days, and let $P(t)$ be the amount remaining at time $t$.
(a) Give the formula for $P(t)$.
(b) Give the differential equation satisfied by $P(t)$.
(c) How much will remain after 5 days?
(d) What is the half-life of this radioactive substance?

Joseph Liao

Numerade Educator

Problem 17

The decay constant for the radioactive element cesium 137 is .023 when time is measured in years. Find its half-life.

Joseph Liao

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Problem 18

Radioactive cobalt 60 has a half-life of $5.3$ years. Find its decay constant.

Joseph Liao

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Problem 19

A sample of radioactive material disintegrates from 5 to 2 grams in 100 days. After how many days will just 1 gram remain?

Steven Clarke

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Problem 20

Ten grams of a radioactive material disintegrates to 3 grams in 5 years. What is the half-life of the radioactive material?

Joseph Liao

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Problem 21

In an animal hospital, 8 units of a sulfate were injected into a dog. After $50 \mathrm{~min}-$ utes, only 4 units remained in the dog. Let $f(t)$ be the amount of sulfate present after $t$ minutes. At any time, the rate of change of $f(t)$ is proportional to the value of $f(t)$. Find the formula for $f(t)$.

Joseph Liao

Numerade Educator

Problem 22

Forty grams of a certain radioactive material disintegrates to 16 grams in 220 years. How much of this material is left after 300 years?

Joseph Liao

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Problem 23

A sample of radioactive material decays over time (measured in hours) with decay constant .2. The graph of the exponential function $y=P(t)$ in Fig. 7 gives the number of grams remaining after $t$ hours. [Hint: In parts (c) and (d) use the differential equation satisfied by $P(t) .$
(a) How much was remaining after 1 hour?
(b) Approximate the half-life of the material.
(c) How fast was the sample decaying after 6 hours?
(d) When was the sample decaying at the rate of .4 gram per hour?

Joseph Liao

Numerade Educator

Problem 24

A sample of radioactive material has decay constant $.25$, where time is measured in hours. How fast will the sample be disintegrating when the sample size is 8 grams? For what sample size will the sample size be decreasing at the rate of 2 grams per day?

Joseph Liao

Numerade Educator

Problem 25

In 1947, a cave with beautiful prehistoric wall paintings was discovered in Lascaux, France. Some charcoal found in the cave contained $20 \%$ of the ${ }^{14} \mathrm{C}$ expected in living trees. How old are the Lascaux cave paintings? (Recall that the decay constant for ${ }^{14} \mathrm{C}$ is .00012.)

Joseph Liao

Numerade Educator

Problem 26

According to legend, in the fifth century King Arthur and his knights sat at a huge round table. A round table alleged to have belonged to King Arthur was found at Winchester Castle in England. In 1976, carbon dating revealed the amount of radiocarbon in the table to be $91 \%$ of the radiocarbon present in living wood. Could the table possibly have belonged to King Arthur? Why? (Recall that the decay constant for ${ }^{14} \mathrm{C}$ is $.00012 .$ )

Joseph Liao

Numerade Educator

Problem 27

A 4500-year-old wooden chest was found in the tomb of the twenty-fifth century B.C. Chaldean king Meskalumdug of Ur. What percentage of the original ${ }^{14} \mathrm{C}$ would you expect to find in the wooden chest?

Joseph Liao

Numerade Educator

Problem 28

In 1938, sandals woven from strands of tree bark were found in Fort Rock Creek Cave in Oregon. The bark contained $34 \%$ of the level of ${ }^{14} \mathrm{C}$ found in living bark. Approximately how old were the sandals? [Note: This discovery by University of Oregon anthropologist Luther Cressman forced scientists to double their estimate of how long ago people came to the Pacific Northwest.]

Joseph Liao

Numerade Educator

Problem 29

Many scientists believe there have been four ice ages in the past 1 million years. Before the technique of carbon dating was known, geologists erroneously believed that the retreat of the Fourth Ice Age began about 25,000 years ago. In 1950, logs from ancient spruce trees were found under glacial debris near Two Creeks, Wisconsin. Geologists determined that these trees had been crushed by the advance of ice during the Fourth Ice Age. Wood from the spruce trees contained $27 \%$ of the level of ${ }^{14} \mathrm{C}$ found in living trees. Approximately how long ago did the Fourth Ice Age actually occur?

Joseph Liao

Numerade Educator

Problem 30

Let $T$ be the time constant of the curve $y=C e^{-\lambda t}$ as defined in Fig. $5 .$ Show that $T=1 / \lambda$. [Hint: Express the slope of the tangent line in Fig. 5 in terms of $C$ and $T$. Then, set this slope equal to the slope of the curve $y=C e^{-\lambda t}$ at $\left.t=0 .\right]$

Joseph Liao

Numerade Educator

Problem 31

Differential Equation and Decay The amount in grams of a certain radioactive material present after $t$ years is given by the function $P(t)$. Match each of the following answers with its corresponding question.
Answers
a. Solve $P(t)=.5 P(0)$ for $t$.
b. Solve $P(t)=.5$ for $t$.
c. $P(.5)$
d. $P^{\prime}(.5)$
e. $P(0)$
f. Solve $P^{\prime}(t)=-.5$ for $t$.
g. $y^{\prime}=k y$
h. $P_{0} e^{k t}, k<0$
Questions
A. Give a differential equation satisfied by $P(t)$.
B. How fast will the radioactive material be disintegrating in $\frac{1}{2}$ year?
C. Give the general form of the function $P(t)$.
D. Find the half-life of the radioactive material.
E. How many grams of the material will remain after $\frac{1}{2}$ year?
F. When will the radioactive material be disintegrating at the rate of $\frac{1}{2}$ gram per year?
G. When will there be $\frac{1}{2}$ gram remaining?
$\mathbf{H}$. How much radioactive material was present initially?

Joseph Liao

Numerade Educator

Problem 32

Consider an exponential decay function $P(t)=P_{0} e^{-\lambda t}$, and let $T$ denote its time constant. Show that, at $t=T$, the function $P(t)$ decays to about one-third of its initial size. Conclude that the time constant is always larger than the half-life.

Joseph Liao

Numerade Educator

Problem 33

Suppose that the function $P(t)$ satisfies the differential equation
$$
y^{\prime}(t)=-.5 y(t), \quad y(0)=10
$$
(a) Find an equation of the tangent line to the graph of $y=P(t)$ at $t=0 .$ [Hint: What are $P^{\prime}(0)$ and $\left.P(0) ?\right]$
(b) Find $P(t)$.
(c) What is the time constant of the decay curve $y=P(t) ?$

Joseph Liao

Numerade Educator

Problem 34

Consider the exponential decay function $y=$ $P_{0} e^{-\lambda t}$, with time constant $T$. We define the time to finish to be the time it takes for the function to decay to about $1 \%$ of its initial value $P_{0}$. Show that the time to finish is
about four times the time constant $T$.

Joseph Liao

Numerade Educator