MTH 112

Section 3.5Exponential Growth & Decay

Modeling Data

Overview

• In this section we apply the concepts of exponential and logarithmic functions to population growth, half-life, and carbon dating.

Population Growth

• The exponential model

Describes the population, A, of a country t years after a starting year t0. A0 is the population in year t0, and k is the growth constant.

kteAA 0

Some Data

YearCity of

TuscaloosaCounty of

Tuscaloosa

1980 75200 137500

1990 77800 150500

2000 77900 164900

2010 90500 194700

Source: Wikipedia. Figures rounded to nearest hundred.

Some Questions

1. What is the growth rate of the City of Tuscaloosa? Use 1980 and 2010 as the years of interest.

2. What is the growth rate of the County of Tuscaloosa? Use 1980 and 2010 as the years of interest.

3. When will the population in the City of Tuscaloosa reach 100000, according to the model?

4. What will the population in the County of Tuscaloosa be in 2020, according to the model?

More Examples

• If the population of a country was 58.3 million in 2006, and is projected to be 54.4 million in 2030, find the growth (or decay) constant, k.

• If the population of a county was 93.8 million in 2004 and the projected growth rate, k, is 0.00185, find the projected population in 2023.

Carbon Dating

• Very similar to population growth, except that k is negative.

• When the amount of carbon-14 present is given as a percentage, then A/A0 is equal to that percentage (expressed as a decimal).

Half-Life

• The half-life of a radioactive element is the length of time required for that element to lose half of its radioactivity.

• For example, if an element has a half-life of 20 years, then if it has 100 grams in 2014, in 2034 it will have 50 grams, in 2054 it will have 25 grams, in 2074 it will have 12.5 grams, and so on.

More About Half-Life

• When k is given, use the following formula to find t:

kt

21

ln

More About Half-Life

• When t is given, use the following formula to find k:

tk

21

ln

Examples

• A certain element has a decay rate, k, of 8.1% per year (k = -.081). Find the half-life of the element.

• A certain element has a half-life of 3485 years. Find the decay rate, k.

• Prehistoric cave paintings were discovered in a cave in France. The paint contained 8% of the original carbon-14. If k = -.000121, estimate the age of the paintings.