Exponential Growth & Decay Steps with Example

by Prof Dr. S. Gramlich

2/28/19

EXPONENTIAL GROWTH AND DECAY STEPS WITH EXAMPLES

in conjunction with "Alg & Trig," 5th ed, by Blitzer, similar to section 4.5, #13
EXAMPLE1

Complete the table for the Population Growth Model for a certain country.

2007 Population

(millions)

Projected 2044 Population

(millions)

Projected Growth Rate,

33.2

19.7

1st DERIVE a formula for k:

A = Ao * e^(kt)

A/Ao =e^(kt) {divide both sides by Ao}

ln(A/Ao) =ln [e^(kt)] {take ln of both sides}

ln(A/Ao) =kt *ln e {log power rule & ln e =1}

ln(A/Ao) =kt *1 {divide both sides by t}

[ln(A/Ao)]/t = k

SUBSTITUTE given values for A, Ao, & t:

[ln(19.7/33.2)]/(2044-2007) =k

[ln(0.5933...)]/37 =

-0.5219.../37=

-0.0141 ~=k {approximately}

(intermediate calculations above are all unrounded using a calculator except the answer which is rounded to 4 decimal places)

Therefore, the projected growth rate for the exponential growth model for the population of the country is approximately -0.0141. Since k is negative, it is referred to as the exponential DECAY rate.

EXAMPLE2

2006 Population

(millions)

Projected 2034 Population

(millions)

Projected Growth Rate,

29.8

10.6

SUBSTITUTE given values for A, Ao, & t:

k = [ln(A/Ao)]/t

k = [ln(10.6/29.8)]/(2034-2006)

= [ln(0.3557...)]/28

= -1.0336.../28

k ~= -0.0369

Therefore, the projected growth rate for the exponential growth model for the population of the country is approximately -0.0369. Since k is negative, it is referred to as the exponential DECAY rate.