College Algebra 9th txtbk

342 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
where A is the number of bacteria present after t hours and A 0 is the number of bacteria
present at t 0. If we start with 1 bacterium, how many bacteria will be present in
(A) 5 hours?
(B) 12 hours?
Calculate the answers to three significant digits.
SOLUTIONS
(A) Use A 0 1 and t 5:
e 1.386(5)
1,020
(B) Use A 0 1 and t 12:
A A 0 e 1.386t Let A 0 1 and t 5.
Calculate to three significant digits.
A A 0 e 1.386t Let A 0 1 and t 12.
e 1.386(12)
16,700,000
Calculate to three significant digits.
MATCHED PROBLEM 2
Repeat Example 2 if A A 0 e 0.783t and all other information remains the same.
Exponential functions can also be used to model radioactive decay, which is sometimes
referred to as negative growth. Radioactive materials are used extensively in medical diagnosis
and therapy, as power sources in satellites, and as power sources in many countries.
If we start with an amount A 0 of a particular radioactive substance, the amount declines
exponentially over time. The rate of decay varies depending on the particular radioactive
substance. A convenient and easily understood measure of the rate of decay is the half-life
of the material—that is, the time it takes for half of a particular material to decay. We can
use the following half-life decay model:
A A 0 ( 1 2) t h
A 0 2 t h
where
A Amount at time t
A 0 Amount at time t 0
h Half-life
Note that when the amount of time passed is equal to the half-life (t h),
A A 0 2 h h A 0 2 1 A 0 1 2
and the amount of radioactive material is half the original amount, as it should be.
EXAMPLE 3 Radioactive Decay
The radioactive isotope gallium 67 ( 67 Ga), used in the diagnosis of malignant tumors, has
a biological half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how
many milligrams will be left after
(A) 24 hours?
(B) 1 week?
Calculate answers to three significant digits.