Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
Exponential Growth and Decay
Use f(t) = Ne
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
A population exhibiting exponential growth, or exponential decay, is modeled
using the function f(t) = Ne
. 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
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
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.
observed only for
the rst 5 hours,
so t can’t be higher
than 5; t can’t be
below 0 because the
at time t = 0.
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