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

.

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