(b)Approximate the time it would take for the initial investment to triple its original
amount. Round to the nearest tenth.
We must find the value of tthat will cause Ato be
Continuous compounding formula
Let
Divide by 1000.
Take natural logarithms.
InDivide by 0.03.Use a calculator.It would take about 36.6 yr for the original investment to triple. NOW TRY
OBJECTIVE 4 Solve applications involving base e exponential growth
and decay.When situations involve growth or decay of a population, the amount or
number of some quantity present at time tcan be approximated by
In this equation, is the amount or number present at time and kis a constant.
The continuous compounding of money is an example of exponential growth. In
Example 9,we investigate exponential decay.
Solving an Application Involving Exponential DecayCarbon 14 is a radioactive form of carbon that is found in all living plants and ani-
mals. After a plant or animal dies, the radioactive carbon 14 disintegrates according
to the function defined by
where tis time in years, yis the amount of the sample at time t, and is the initial
amount present at
(a)If an initial sample contains g of carbon 14, how many grams, to the
nearest tenth, will be present after 3000 yr?
Let and in the formula, and use a calculator.
g
(b)About how long would it take for the initial sample to decay to half of its original
amount? (This is called the half-life.) Round to the nearest unit.
Let and solve for t.
Substitute inDivide by 10.Take natural logarithms; InInterchange sides. Divide byUse a calculator.The half-life is about 5728 yr. NOW TRY
tL 5728
t= -0.000121.
ln^12
- 0.000121
ln ek=k.
1
2
=-0.000121t
1
2
= e-0.000121t
5 = 10 e-0.000121t y=y 0 ekt.
y=^12 1102 = 5,
y= 10 e-0.000121^130002 L6.96
y 0 = 10 t= 3000
y 0 = 10
t=0.
y 0
y=y 0 e-0.000121t,
EXAMPLE 9
y 0 t= 0
yy 0 ekt.
tL36.6
t =
ln 3
0.03
ln 3=0.03t ek=k
ln 3=ln e0.03t
3 =e0.03t
3000 = 1000 e0.03t A= 3 P=3000, P=1000, r=0.03.
A=Pert
31 $1000 2 =$3000.
618 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions
NOW TRY
EXERCISE 8
Suppose that $4000 is invested
at 3% interest for 2 yr.
(a)How much will the
investment grow to if it is
compounded continuously?
(b)Approximate the time it
would take for the amount
to double. Round to the
nearest tenth.
NOW TRY ANSWERS
- (a)$4247.35 (b)23.1 yr
- (a)4.2 g (b) 1612 yr
NOW TRY
EXERCISE 9
Radium 226 decays according
to the function defined by
where tis time in years.
(a)If an initial sample contains
g of radium
226, how many grams, to
the nearest tenth, will be
present after 150 yr?
(b)Approximate the half-life
of radium 226. Round to
the nearest unit.
y 0 =4.5y=y 0 e-0.00043t,