13 - using exponential and logarithmic functions

8/16/2019 13 - Using Exponential and Logarithmic Functions

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7-8 Study Guide and InterventionUsing Exponential and Logarithmic Functions

Exponential Growth and Decay

ExponentialGrowth

f ( x ) = a ekt where a is the initial value of y , t is time in ears, an! k is a "onstant

re#resentin$ the rate of continuous growth.

ExponentialDecay

f ( x ) = a e−kt where a is the initial value of y , t is time in ears, an! k is a "onstant

re#resentin$ the rate of continuous decay.

Example: POPULATION In 2000 the world pop!lation wa" e"timated to #e $%&2' #illion people%In 200( it wa" $%(&( #illion%

a% Determine the )al!e o* k the world+" relati)e rate o* ,rowth% the )al!e o* k the world+" relati)e rate o* ,rowth%

y = a ekt

%ormula for "ontinuous $rowth

6.515 = 6.124 ek

(5) y = &', a = &'*+, an! t = * - * or

6.515

6.124 = e

5k Divi!e ea"h si!e . &'*+'

ln6.515

6.124 = ln e

5k Pro#ert of E/ualit for 0o$arithmi" %un"tions

ln6.515

6.124 = 5k ln e

x= x

0.01238 ≈ k Divi!e ea"h si!e . an! use a "al"ulator'

The world’s relative rate of growth is about 0.01238 or 1.2

#% -hen will the world+" pop!lation reach .%( #illion people/

!.5 ≈ 6.124e0.01238t

y = 1', a = &'*+, an! k = '*23

7.5

6.124 ≈ e

0.01238t Divi!e ea"h si!e . &'*+'

ln7.5

6.124 ≈ ln e

0.01238t Pro#ert of E/ualit for 0o$arithmi" %un"tions

ln7.5

6.124 ≈ 0.01238t ln e

x = x

16.3!22 ≈ t Divi!e ea"h si!e . '*23 an! use a "al"ulator'

The world’s "o"ulation will rea#h !.5 billion in 2016.

Exerci"e

&% A1ON DATING $se the for%ula y = ae−0.00012t & where a is the initial a%ount of #arbon 14& t is the nu%ber o

'ears ago the ani%al lived& and y is the re%aining a%ount after t 'ears.

a% (ow old is a fossil that has lost )5 of its *arbon+14,

4ha#ter 1 53 Glencoe Algebra


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7-8 Study Guide and Intervention (continued)Using Exponential and Logarithmic Functions

Lo,i"tic Growth logisti# fun#tion %odels the /+#urve of growth of so%e set . The initial stage of growth isa""roi%atel' e"onential then& as saturation begins& the growth slows& and at so%e "oint& growth sto"s.

Example: The pop!lation o* a certain "pecie" o* *i"h in a la3e a*ter t year" i" ,i)en #y P 4t 5 61880

1+1.42 e−0.037t .

a% Graph the *!nction%

#% 7ind the hori8ontal a"ymptote% -hat doe" it repre"ent in the "it!ation/

The horiontal as'%"tote is P t = 1880. The "o"ulation of fish will rea#h a #eiling of 1880.

c% -hen will the pop!lation reach &9.(/

18!5 =1880

1+1.42e−0.037t Re#la"e P (t ) with 31'

18!5 1+1.42e−0.037 t = 1880 Multi#l ea"h si!e . ( 1+1.42e−0.037 t )'

2662.5 e−0.037 t = 5 5im#lif an! su.tra"t 31 from ea"h si!e'

e−0.037t =

5

2662.5Divi!e ea"h si!e . *&&*''

0.03!t = ln5

2662.5Ta6e the natural lo$arithm of ea"h si!e'

t = ln ( 52662.5 ) 7 0.03! Divi!e ea"h si!e . - '21't ≈ 16).66 7se a "al"ulator'

The "o"ulation will rea#h 18!5 in about 1!0 'ears.

Exerci"e

&% ssu%e the "o"ulation of gnats in a s"e#ifi# habitat follows the fun#tion

P t =17,000

(1+15e−0.0082 t ).

a% ra"h the fun#tion for t 9 0.

#% :hat is the horiontal as'%"tote,



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c% :hat is the %ai%u% "o"ulation,

d% :hen does the "o"ulation rea#h 15&000,


13 - using exponential and logarithmic functions

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