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Exponential
Growth
Subject
Area
This activity
has been designed for the Senior Mathematics curriculum. Specifically,
it forms part of the MTB 3G1 curriculum after the section on Compound
Interest, MTT 3G1 after the section on Compound Interest, MAT
3A1 after the section on Compound Interest, MTB 4G1 after the
section on Compound Interest, and MTT 4G1 after the section on
Exponential Growth.
Learning
Outcomes
Teaching,
learning and evaluation will focus on the student's ability to:
- Review
and consolidate skills in exponential calculations;
- Become
aware of the rate of growth of the world's population, and the
implications of that continued growth;
- Become
aware of the relationships among population growth, gross domestic
product, inflation rate, and population density;
- Learn
applications and implications of exponential growth.
Classroom
Development
- In MTT
4G1 teachers should review using exponential growth in compound
interest questions.
- Teachers
should introduce the topic using the examples given. It will
be necessary to define some terminology such as GDP, arable,
and per capita.
- Students
could work cooperatively in small groups solving the questions
for the countries described. 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,
and 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.
Resources
Samuelson,
Robert J. Book of Vital Word Statistics, The Economist Books.
New York: Times Books, Random House, 1990.
Student
Activities
Exponential
Growth
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 |
$1.76 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 |
| (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 |
| (iv) 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?
Solution
Let the
number of years required be n.
50 = 26(1.009)n
(1.009)n
= 50 /26= 1.9230769
Use your
calculator to estimate n.
| n |
(1.009)n |
| 50 |
1.565 |
| 100 |
2.449 |
| 75 |
1.958 |
| 70 |
1.872 |
| 73 |
1.923 |
n = 73 years
Therefore,
Canada's popluation will be 50 million people in the year 2061.
The table
below shows various statistics for Brazil, Phillipines, and the
United States in 1988, and annual real growth rates. Use the information
in this table to answer the questions that follow.
| Item |
Brazil |
Philippines |
United States |
| Population |
144.4 million |
58.7 million |
246.3 million |
| Annual Growth Rate |
+2.2% |
+2.4% |
+1.0% |
| GDP per capita |
$2451 |
$662 |
$19815 |
| Annual Growth Rate |
+2.4% |
+1.3% |
+3.0% |
| Price per litre of milk |
$0.41 |
$1.16 |
$0.72 |
| Price per kg of bread |
$0.69 |
$1.01 |
$2.97 |
| Annual Inflation Rate |
+390.2% |
+9.2% |
+3.6% |
- a) Calculate
the population and GDP per capita for each country in the year
2008.
- Brazil
- Philippines
- United
States
-
- b)
In Brazil in 1988, the average person spent approximately
$700 on food. Use the inflation rate to calculate the cost
of this food in the year 1994. Also, calculate the GDP per
capita in 1994. Compare the two figures.
- 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 |
| Haiti |
+1.5% |
-).8% |
|
| Hong Kong |
+1.2% |
+7.6% |
|
| Pakistan |
+3.1% |
+6.3% |
|
| Tanzania |
+2.9% |
+2.0% |
|
| Former West Germany |
-0.1% |
+1.7% |
|
- At the
present growth rate, when will the population of the Philippines
reach100 million people?
- The area
of the United States is 9 373 000 km2. Calculate the population
density in 1988, and in 2025.
- The area
of Brazil is 8 456 510 km2. Of this area, 9.2% is arable.
- (a)
Calculate the area of arable land.
- (b)
Calculate the population density per square kilometre of
arable land in 1988, and in 2025.
- The area
of Canada is 9 222 970 km2. Of this area, 5.0% is arable.
- (a)
Calculate the area of arable land.
- (b)
Calculate the population density per square kilometre of
arable land in 1988, and in 2025.
- Calculate
the price of a litre of milk in the Philippines in 2020.
- When
will the price of a kilogram of bread be $10.00 in:
- The
United States of America?
- Brazil?
- The
Philippines?
- In the
Philippines, 37% of the land area is forest. The land is being
deforested at the rate of 2.7% 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 50%?
- (a) In
what year will the population of the Philippines be the same
as the population of the United States?
- (b)
Calculate the population densities of both the United States
and the Philippines in that year. The area of the Philippines
is 300 000 km2 and the area of the United States is 9 373
000 km2.
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