Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
433
Exponential Growth and Decay
Use f(t) = Ne
kt
to measure things like population
Note: Problems 19.36–19.39 refer to the laboratory experiment described here.
19.36 A group of scientists conduct an experiment to determine the effectiveness
of an agent designed to accelerate the growth of a specific bacterium. They
introduce 40 bacterial colonies into a growth medium, and exactly two hours
later the number of colonies has grown to 200. In fact, the number of colonies
grows exponentially for the first 5 hours of the experiment.
Design a function f(t) that models the number of bacterial colonies t hours
after the experiment has begun (assuming 0 ≤ t ≤ 5) that includes constant of
proportionality k.
A population exhibiting exponential growth, or exponential decay, is modeled
using the function f(t) = Ne
kt
. The variables in the formula are defined as
follows: N = original population (in this case N = 40 colonies), t = number of
hours elapsed, and k = the constant of proportionality.
f(t) = 40e
kt
Note: Problems 19.36–19.39 refer to the laboratory experiment described in Problem 19.36.
19.37 Calculate k.
Exactly t = 2 hours after the start of the experiment, the number of bacterial
colonies was f(2) = 200. Substitute t = 2 and f(2) = 200 into the population
model f(t) = 40e
kt
from Problem 19.36.
Isolate the exponential expression on the right side of the equal sign.
Solve for k by taking the natural logarithm of both sides of the equation.
Exponential
growth is
observed only for
the rst 5 hours,
so t can’t be higher
than 5; t can’t be
below 0 because the
experiment begins
at time t = 0.
Don’t try
to guess what k
is or plug a number
from the original
problem in for k. You
almost always have to
calculate k based on
the given information,
which is exactly
what you do in
Problem 19.37.
Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
434
Note: Problems 19.36–19.39 refer to a laboratory experiment described in Problem 19.36.
19.38 Calculate the number of colonies that were present 5 hours after the
experiment began, rounding the answer to the nearest whole number.
According to Problem 19.37, . Substitute k into the population model
defined in Problem 19.36.
Evaluate f(5).
Use a calculator to calculate the exponent of e: .
After 5 hours, the 40 original colonies have grown to 2,236 colonies.
Note: Problems 19.36–19.39 refer to a laboratory experiment described in Problem 19.36.
19.39 Approximately how many minutes does it take the bacterial colonies to grow in
number from 40 to 120? Round the answer to the nearest whole minute.
According to Problem 19.38, the population is modeled by the function
. Calculate t when f(t) = 120.
Eliminate the natural exponential function by taking the natural logarithm of
each side of the equation.
To calculate
the population 5
hours after the
experiment starts,
substitute t = 5 into
the population model.
Multiply
both sides of
the equation by the
reciprocal of this
number to solve
for t.
Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
435
Convert t ≈ 1.365212389 hours into minutes.
60(1.365212389) ≈ 81.91274334
There were a total of 120 bacterial colonies approximately 82 minutes after the
experiment began.
Note: Problems 19.40–19.43 refer to the radioactive decay of carbon-14.
19.40 Upon the death of an organism, the amount of carbon-14 present in that
organism decays exponentially, with a half-life of approximately 5,730 years.
Construct a function g(t) that models the amount of carbon-14 present in an
organism t years after its demise, including the constant of proportionality
accurate to eight decimal places.
Apply the exponential growth and decay formula g(t) = Ne
kt
, such that N = the
original amount of carbon-14 in the living organism. According to the problem,
carbon-14 has a half-life of 5,730 years. Therefore, when t = 5,730, .
Substitute t and g(t) into the function and solve for k.
Multiply both sides of the equation by to isolate the exponential expression.
Take the natural logarithm of both sides of the equation to isolate k.
Substitute k into g(t).
g(t) = Ne
–0.00012097t
There are
60 minutes in an
hour, so multiply t by
60 to calculate how
many minutes are in
1.365212389 hours.
Half-life
is the amount
of time it takes
something to decay
to half of its original
amount. Carbon-14
has a half-life of
5,730 years, so a 100
kg sample of carbon-
14 will decay to
100 ÷ 2 = 50 kg in
5,730 years.
The original version of
g(t): g(t) = Ne
kt
.
Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
436
Note: Problems 19.40–19.43 refer to the radioactive decay of carbon-14, as modeled by the
function g(t) generated in Problem 19.40.
19.41 Approximately how long does it take N grams of carbon-14 to decay by two-
thirds? Report the answer rounded to the nearest whole year.
After some time t has elapsed, . Substitute this value into
g(t) = Ne
–0.00012097t
(the half-life model from Problem 19.40) and solve for t.
It takes approximately 9,082 years for a mass of carbon-14 to decay by two-thirds.
Note: Problems 19.40–19.43 refer to the radioactive decay of carbon-14, as modeled by the
function g(t) generated in Problem 19.40.
19.42 Approximately how long does it take N grams of carbon-14 to decay by three-
fourths?
The half-life of carbon-14 is approximately 5,730 years. Therefore, in 5,730
years, a mass N of carbon-14 will decay to a mass of . In an additional 5,730
years, the mass will again decrease by half: . Thus, one-fourth of the
mass is left after 2(5,730) = 11,460 years.
If the mass
decreased by
two-thirds, that
means one-third of
the original mass
N is left.
Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
437
Note: Problems 19.40–19.43 refer to the radioactive decay of carbon-14, as modeled by the
function g(t) generated in Problem 19.40.
19.43 If an organism contains 15 grams of carbon-14 at the time of its death, how
much carbon-14 will the organism contain 100 years later? Round the answer
to the thousandths place.
Substitute N = 15 and t = 100 into the half-life model g(t) defined in Problem
19.40 and evaluate g(100).
The organism will contain approximately 14.820 grams of carbon-14 one
hundred years after its death.
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