Population Growth
and Decay
College Algebra:
One Step at a Time. Page 539:
#15, 20, 21, 23, 26, 27 and MORE!!
Dr. Robert J. Rapalje
Seminole State College of Florida
Altamonte Springs Campus
To see
Section 4.05, with
detailed explanations, examples, exercises, and answers,
click here!
Solving equations
using logarithms is not nearly as hard as you think it is, especially if it
is explained in living color! Consider this exercise from page 523.
Notice how the colors make the exercises so much easier to follow. Can
you imagine what this would look like in black and white? Most of our
television is in color--why not math? Have your graphing calculator
ready!
P. 535 #
15.
The amount of radioactive substance present at
any time is given by 
,
where 
is
the initial amount
and t is the
time in years. If
there are 25 grams
of the substance left after 2000 years,
how much was there originally?
Solution:
,
and is
the unknown!
Find ,
by dividing both sides of the equation by
:
or
approximately 55.6 grams.
P. 538 #
20, 21, 23.
20. The
population of a city was 40,000 in the year 1990. In 1995, the population
of the city was 50,000. Find the value of k in the formula
.
Solution:
and
First
find k, by dividing both sides of the equation by 40,000:
When
calculating this, I strongly recommend that you leave the result in
fractional form. Do NOT calculate the decimal value. If you write down a
decimal value, you may be rounding off the answer, which could lead
to an incredible error in the final answer. It’s okay to round off the
final answer, but NEVER round off a number and then use that number in
subsequent calculations!
[NOTE: In this case,
if you calculate the decimal value, the result is 1.25, which is EXACT! No
rounding takes place, so there is NO round off error. In this case, using
1.25 is equally correct! However, don’t do this in the rest of the
exercises!]
Now, take the ln of each side of the equation to
“undo” the “e^ “:
Divide
both sides by 5:
The calculator needs
parentheses: ≈
0.0446287
. . .
You
may want to store this number in your calculator using the [STO→] button.
For TI 83/84 press [STO→] [ALPHA] [k].
For TI 85/86 , after the [STO→] , the calculator is automatically in the
alpha mode so you do NOT press the [ALPHA] key.
21. Now,
use the value of k
obtained in #20 to find the expected population in
2005. Notice that since 
means
that t=0 in
1990, in 2005, t=15.
or
To
use your calculator do this, having already entered the value of
k in the
calculator, press [40000]
[2nd] [ln] [ your calculator may or may
not automatically open parentheses] [15]
[ALPHA] [k]
[closed parentheses].
y
≈ 78,125.
23. Use
the value of k
obtained in #20 to determine how long it will take the
population to double. If the population doubles, where
was
the initial population, the new population will be
,
and you can substitute into the equation as follows:
Divide both sides by
:
Take the ln
of each side:
To solve for t, divide
both sides by k.
If you have the value of k in the calculator already from the previous
exercises, just divide 
by
[alpha] [k]
.
t =15.5314186,
or t= 15.5 years
P. 539 # 26 – 27, and MORE!
26.
The population
of a city was 85,000
in the year
2000, and 88,000
in 2002. At this rate
of growth, what population should be expected in 2005?
Solution:
and
First find k, by dividing both
sides of the equation by 85,000:
When calculating
this, I strongly recommend that you leave the result in fractional form. Do
NOT calculate the decimal value. If you write down a decimal value, you are
rounding off the answer, which could lead to an incredible error in the
final answer. It’s okay to round off the final answer, but NEVER round off
a number and then use that number in subsequent calculations!
Now, take the ln of each side
of the equation to “undo” the “e^ “:
Divide both sides by
2:
The calculator needs
parentheses: 
≈ 0.017342779 . . .
You may want to
store this number in your calculator using the [STO→] button. For TI 83/84
press [STO→] [ALPHA] [k]. For TI 85/86 , after the [STO→] , the
calculator is automatically in the alpha mode so you do NOT press the
[ALPHA] key.
Now, use that value
of k to answer the question: “Find y when t=5.”
or 
To use your calculator do this,
having already entered the value of k in the calculator, press [85000] [2nd]
[ln] [ your calculator may automatically open parentheses] [5]
[ALPHA] [k] [closed parentheses].
y ≈
92699.69 , which rounds off to 92,700. Final Answer!
27.
At this rate of growth, how long will
the population in the city in the previous exercise to double?
Solution:
When the
population doubles, whatever
is,
.
, where k is the value in the calculator!
First find k, by
dividing both sides of the equation by
:
Take the ln of
each side of the equation to “undo” the “e^ “:
Now, since you are solving for t,
you must divide both sides by k:
, where k is the value (0.017342779 . . .
) in your calculator!
t
≈ 39.867 years (approximately 40 years!)
EXTRA NEW
EXERCISES--Not in the book!!!
28.
How long will it take the population from
# 26
to reach 200,000?
Solution:
, 
, and
k
is in the calculator.
Solve for
t,
by dividing both sides of the equation by
85,000:
Divide both sides by k, the value
(0.017342779 . . .
)
in
the calculator. The answer is t.
≈ 49.338 ≈ 49
years. Final Answer
It
might be interesting to look at the graph of this population function,
contributed by Lane Vosbury: ,
Seminole State Mathematics Department Chairman
29.
The
population of a city was 184,000 in the year 1996, and 310,000 in the year
2003. At this rate of growth, find the value of k, and estimate the
population that should be expected
a.
in 2008? b. in
2010? c. in 2015?
Solution:
and
First find k, by dividing both sides of the equation
by 184,000:
Remember, it’s better to leave
the result in fractional form. If you calculate the decimal value, you will
be tempted to round off the answer, which could lead to an incredible error
in the final answer. As I said before, it’s okay to round off the final
answer, but NEVER round off a number and then use that number in subsequent
calculations!
Now, take the ln of each side
of the equation to “undo” the “e^ “:
Divide both sides by 7:
The calculator needs
parentheses:
≈ 0.0745195057
. . .
You may want
to store this number in your calculator using the [STO→] button. For TI
83/84 press [STO→] [ALPHA] [k]. For TI 85/86 , after the [STO→] , the
calculator is automatically in the alpha mode so you do NOT press the
[ALPHA] key.
a) Now,
use that value of k
(0.0745195057
. . .
) to find y
in 2008. In 2008,
t=12.
or 
To use your
calculator do this, having already entered the value of
k in the
calculator, press [184000]
[2nd] [ln] [ your calculator may or may not automatically
open parentheses] [12]
[ALPHA] [k]
[closed parentheses].
y ≈
449,965.0108
, which rounds off to
450,000.
b) Now, use that value
of k
to find y
in 2010. In 2008,
t=14.
or 
To use your calculator do this,
having already entered the value of k
in the calculator, press [184000]
[2nd] [ln] [ your calculator may or may not automatically
open parentheses] [14]
[ALPHA] [k]
[closed parentheses].
y
≈ 522,282.6087
, which rounds off to
522,000.
c)
Now, use that value of k
to find y
in 2015. In 2015,
t=19.
or
To use your
calculator do this, having already entered the value of
k (0.0745195057
. . .
) in the
calculator, press [184000]
[2nd] [ln] [ your calculator may or may not automatically
open parentheses] [19]
[ALPHA] [k]
[closed parentheses].
y ≈
758,093.2247
, which rounds off to
758,000.
30. How
long will it take the population in the previous exercise
a.
to
double? b.
to triple?
Solution:
a.
When the population doubles,
whatever
is,
.
, where k (0.0745195057
. . .
) is the value in the
calculator!
First find k,
by dividing both sides of the equation by
:
Take the ln of each side of the equation
to “undo” the “e^ “:
Now, since you are solving for
t, you must
divide both sides by k:
, where
k (0.0745195057
. . .
) is the value
in your calculator!
t
≈ 9.30155
years (approximately
9.3 years!)
b. When the population
triples, whatever
is,
.
, where k is the value in the
calculator!
First find k, by dividing both sides of the equation
by :
Take the ln of each side of the equation
to “undo” the “e^ “:
Now, since you are solving for
t, you must
divide both sides by k:
, where
k is the value
in your calculator (0.0745195057
. . .
)!
t
≈ 14.7426 years
(approximately 14.7
years!)
31. How
many years will it take this population reach
500,000?
Solution:
, 
, and k
is in the calculator (0.0745195057
. . .
).
Solve for t,
by dividing both sides of the equation by
184,000:
Divide both sides by k,
the value in the calculator. The answer is
t.
≈ 13.4149 ≈
13.4 years. Final
Answer!
32.
How many years will it take this
population to reach 1,000,000?
Solution:
, 
, and k
is in the calculator.
Solve for t,
by dividing both sides of the equation by
184,000:
Divide both sides by
k, the value in the calculator. The
answer is t.
≈ 22.71646 ≈
22.7 years. Final Answer
[NOTE: As a check, subtract the answers from the last two problems
(#31 and 32) and compare to
#30a) where the population doubles! ]
(Notice that 22.7 − 13.4 = 9.3 years)
Extra Problems
#1.
A
pharmaceutical company makes a vaccine that grows at a rate of 2.5% per
hour. How many units of this organism must they have initially to have 1000
units after 5 days.
Solution:
In this problem, rate of growth is 2.5%,
so 
.
Also,
.
You must solve for 
using
this formula
or
units
.
#2.
Some
substances, such as carbon 14, have very long half lives and are used by
archaeologists to date fossils, plants and animals fairly accurately. The
half life of carbon 14 is about 5600 years. At an archaeological site, a
thigh bone was found and contained approximately one third of the carbon 14
normally found in a living thigh bone. Calculate the age of the bone to the
nearest year.
Solution:
When 
years,
half of the carbon remains, so
Divide both sides by
Take the ln
of each side:
Divide both sides by 
Now, the question is this: If one-third of the carbon remains
(i.e., if ),
how old is the bone (i.e., solve for t)??
Divide both sides by
.
Take the ln of each
side:
Divide both sides by 
.
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