1. Further Applications and Modeling with
Further Applications and Modeling with Exponential and Logarithmic Functions
Physical Science Applications:
A0 is some initial quantity
t represents time
k > 0 represents the growth constant, and k < 0 represents the decay constant

2. Exponential Decay Function Involving Radioactive Isotopes
Example Nuclear energy derived from radioactive isotopes
can be used to supply power to space vehicles. Suppose that
the output of the radioactive power supply for a certain
satellite is given by the function defined by
where y is measured in watts and t is the time in days.
What is the initial output of the power supply?
After how many days will the output be reduced to
35 watts?
(c) After how many days will the output be half of its initial amount? (That is, what is its half-life?)

3. Exponential Decay Function Involving Radioactive Isotopes
Solution
Let t = 0 and evaluate y.
The initial output is 40 watts.
(b) Let y = 35 and solve for t.

4. Exponential Decay Function Involving Radioactive Isotopes
(c) Because the initial amount is 40, the half-life is the value of t for which

5. Exponential Decay Function Involving Radioactive Isotopes
The half-life can be obtained from the graph of
by noting that when t = x = 173,
y  20 =

6. Age of a Fossil using Carbon-14 Dating
Example Carbon-14 is a radioactive form of carbon found
in all living plants and animals. After a plant or animal dies,
the radiocarbon disintegrates. Scientists determine the age of
the remains by comparing the amount of carbon-14 present
with the amount found in living plants and animals. The
amount of carbon-14 present after t years is given by
Find the half-life.
Solution Let
Divide by A0.

7. Age of a Fossil using Carbon-14 Dating
The half-life is 5700 years.
Take the ln of both sides.
ln ex = x and quotient rule for logarithms
Isolate t.
Distribute and use the fact that ln1 = 0.

8. Finding Half-life Example Radium-226, which decays according to
has a half-life of about 1612 years. Find k. How long does it
take a 10-gram sample to decay to 6 grams?
Solution The half-life tells us that A(1612) = (½)A0.

9. Finding Half-life Thus, radium-226 decays according to the equation
Now let A(t) = 6 and A0 = 10 to
find t.

10. Financial Applications
Example How long will it take $1000 invested at
6%, compounded quarterly, to grow to $2700?
Solution Find t when A = 2700, P = 1000, r = .06,
and n = 4.

11. Amortization Payments
A loan of P dollars at interest i per period may be amortized in n equal periodic payments of
R dollars made at the end of each period, where
The total interest I that will be paid during the term of the loan is

12. Using Amortization to Finance an Automobile
Example You agree to pay $24,000 for a used SUV. After
a down payment of $4000, the balance will be paid off in
36 equal monthly payments at 8.5% interest per year. Find the
amount of each payment. How much interest will you pay
over the life of the loan?
Solution