| |
Applications
of Exponential and Logarithmic Functions
Subject
Area
This activity
has been designed for the Mathematics MAT 4A1 curriculum, after
the section on Exponential and Logarithmic Functions.
Learning
Outcomes
- Teaching,
learning and evaluation will focus on the student's ability
to:
- Review
and consolidate skills in exponential and logarithmic calculations;
- Discover
through the use of mathematical skills the rate of growth of
the world's population and the implications of that continued
growth;
- Interpret
data which express the relationships among population growth,
gross domestic product, inflation rate, and population density;
- Learn
applications and implications of exponential growth.
Classroom
Development
- This
unit should be done after the section on Exponential and Logarithmic
Functions.
- Review
using exponential functions in compound interest questions.
- Introduce
the topic using the examples given. It will be necessary to
define some terminology such as GDP, arable, per capita.
- Students
could work cooperatively in small groups solving the questions
for the countries listed. An alternative would be for the students
to refer to the resource listed and investigate countries of
their choice. The resource contains many additional data items
that could be investigated in the similar ways-some examples
are defense spending, education spending, health care costs,
trade balances.
- Use class
discussion time to see that the mathematics has been done correctly
and to review the skills being used. Discuss the exponential
nature of population growth.
- Spend
some class time to discuss some of the implications of population
growth, gross domestic product, inflation rate, and population
density.
Timing
Allow two
periods for the completion of this activity.
Resource
Samuelson,
Robert J. Book of Vital Word Statistics, The Economist Books,
Times Books. Random House, New York: 1990.
Student
Activities
Applications
of Exponential and Logarithmic Functions
The table
below shows various statistics for Canada in 1988, and annual
real growth rates.
| ITEM |
1988 Data |
Real Growth Rate |
| Population |
26 million |
+0.9% |
| GDP per capita |
$18 834 |
+3.5% |
| Forests |
38% of land area |
+0.6% |
| Price of Milk |
$1.19 per litre |
+4.0% |
| Price of Bread |
$176 per kilogram |
+4.0% |
Example
1
Recalculate
the data for the year 2008.
Solution
| i) Population |
= 26(1.009)20
= 26(1.196254)
= 31.1 million |
| ii) Total GDP in 1988 |
= (26 million people) X ($18 834 per person)
= $489 684 million |
| Total GDP in 2008 |
= 489 684(1.035)20
= $974 368 million |
| GDP per capita in 2008 |
= $974 368 million ÷ 31.1
= $31 330 |
| iii) Price of milk in 2008 |
= 1.19 (1.04)20
= 1.19 (2.1911)
= $2.61 per litre |
| v) Price of bread in 2008 |
= 1.76 (1.04)20
= 1.76 (2.1911)
= $3.86 per kilogram
|
Example
2
In what
year will the population of Canada be 50 million people?
Solution
Let the
number of years required be n.
| 50 |
= 26 (1.009)n |
| log50 |
= log [26 (1.009)n] |
| log50 |
= log 26 + n log 1.009 |
| n |
= log 50 - log 26 / log 1.009 |
| n |
= 72.89 |
| n |
= 73 years |
Therefore,
Canada's human population will be 50 million in 2061.
Activities
The table
below shows various statistics for Mexico, Nigeria, and the United
Kingdom in 1988, and annual real growth rates. Use the information
in this table to answer the questions that follow.
| ITEM |
Mexico |
Nigeria |
United Kingdom |
| Population |
82.73 million |
105.0 million |
57.08 million |
| Annual Growth Rate |
+2.1 % |
+3.3% |
+0.3% |
| GDP per capita |
$2102 |
$287 |
$14 477 |
| Annual Growth Rate |
+1.0% |
-1.3% |
+2.8% |
| Price/L of milk |
$0.67 |
$2.39 |
$0.95 |
| Price/kg of bread |
$1.31 |
$2.46 |
$1.40 |
| Annual Inflation Rate |
+7.8% |
+38.25% |
+5.3% |
- a) Calculate
the population and GDP per capita for each country in the year
2008.
- b)
In Mexico in 1988, the average person spent approximately
$500 on food. Use the inflation rate to calculate the cost
of this food in the year 2008. Compare this figure with
the GDP per capita in 2008.
- c)
Use the information given below to state which of the countries
listed have increasing GDP per capita, and which have decreasing
GDP per capita.
| Country |
Annual Population Growth
Rate |
Annual GDP Growth Rate |
Increasing or Decreasing |
| India |
+2.0% |
+5.7% |
|
| France |
+0.4% |
+1.9% |
|
| Jamaica |
+1.8% |
+1.2% |
|
| Kenya |
+4.9% |
+4.3% |
|
| Sri Lanka |
+1.5% |
+4.2% |
|
- At the
present growth rate, when will the population of Nigeria reach
150 million people?
- The area
of the United Kingdom is 244 000 square km. Calculate the population
density in 1988, in 2000, and in 2050.
- The area
of Nigeria is 910 770 square km.Of this area, 34.4% is arable.
- (a)
Calculate the area of arable land.
- (b)
Calculate the population density per square kilometer of
arable land in 1988 and 2020.
- The area
of Canada is 9 222 970 square km. Of this area, 5.0% is arable.
- (a)
Calculate the area of arable land.
- (b)
Calculate the population density per square kilometer of
arable land in 1988 and 2020.
- Calculate
the price of a litre of milk in Mexico in 2020.
- When
will the price of a kilogram of bread be $10.00?
- (a)
In Mexico?
- (b)
In Nigeria?
- (c)
In the United Kingdom?
- In Mexico,
23% of the land area is forest. The land is being deforested
at the rate of 1.1% per year.
- (a)
What percentage of the land will be forest in 20 years?
- (b)
How many years will it take for the percentage of forested
land to be reduced by 15%?
- In what
year will the price of a litre of milk be the same in both Mexico
and the United Kingdom?
|
|
