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Section 8A Growth: Linear vs. Exponential. Pages 490-495. 8-A. Growth: Linear vs Exponential.

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### Section 8AGrowth: Linear vs. Exponential

Pages 490-495

Growth: Linear vs Exponential

Imagine two communities, Straightown and Powertown, each with an initial population of 10,000 people. Straightown grows at a constant rate of 500 people per year. Powertown grows at a constant rate of 5% per year.

Compare the population growth of Straightown and Powertown.

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Straightown: initially 10,000 people and growing at a rate of 500 people per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Powertown: initially 10,000 people and growing at a rate of 5% per year

Two Basic Growth Patterns

Linear Growth (Decay)occurs when a quantity increases (decreases) by the same absolute amount in each unit of time.

Example: Straightown -- 500 each year

Exponential Growth (Decay)occurs when a quantity increases (decreases) by the same relative amount—that is, by the same percentage—in each unit of time.

Example: Powertown: -- 5% each year

Linear/Exponential Growth/Decay?

The number of students at Wilson High School has increased by 50 in each of the past four years.

- Which kind of growth is this?
- Linear Growth

- If the student populations was 750 four years ago, what is it today?
- 4 years ago: 750
- Now (4 years later): 750 + (4 x 50) = 950

Linear/Exponential Growth/Decay?

The price of milk has been rising with inflation at 3.5% per year.

- Which kind of growth is this?
- Exponential Growth

- If the price was $1.80/gallon two years ago, what is it now?
- 2 years ago: $1.80/gallon
- Now (2 years later): $1.80 × (1.035)2
- = $1.93/gallon

Linear/Exponential Growth/Decay?

Tax law allows you to depreciate the value of your equipment by $200 per year.

- Which kind of growth is this?
- Linear Decay

- If you purchased the equipment three years ago for $1000, what is its depreciated value now?
- 3 years ago: $1000
- Now (3 years later): $1000 – (3 x 200)
- = $400

Linear/Exponential Growth/Decay?

The memory capacity of state-of-the-art computer hard drives is doubling approximately every two years.

- Which kind of growth is this?
- [doubling means increasing by 100%]
- Exponential Growth

- If the company’s top of the line drive holds 300 gigabytes today, what will it hold in six years?
- Now: 300 gigabytes
- 2 years: 600 gigabytes
- 4 years: 1200 gigabytes
- 6 years: 2400 gigabytes

Linear/Exponential Growth/Decay?

The price of DVD recorders has been falling by about 25% per year.

- Which kind of growth is this?
- Exponential Decay

- If the price is $200 today, what can you expect it to be in 2 years?
- Now: $200
- 2 years: 200 x (0.75)2
- = $112.50

More Practice

The population of Danbury is increasing by 505 people per year. If the population is 15,000 today, what will it be in three years?

16,515

During the worst periods of hyper inflation in Brazil, the price of food increased at a rate of 30% per month. If your food bill was $100 one month during this period, what was it two months later?

$169

The price of computer memory is decreasing at a rate of 12% per year. If a memory chip costs $80 today, what will it cost in 2 years?

$61.95

The Impact of Doubling

Parable 1 From Hero to Headless in 64 Easy Steps

Parable 2 The Magic Penny

Parable 3 Bacteria in a Bottle

Parable 1

From Hero to Headless in 64 Easy Steps

Parable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

Parable 1

From Hero to Headless in 64 Easy Steps

Parable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

264 – 1 = 1.8×1019 =

≈ 18 billion, billion grains of wheat

This is more than all the grains of wheat harvested in human history.

The king never finished paying the inventor and according to legend, instead had him beheaded.

Parable 2

The Magic Penny

Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies.

Parable 2

The Magic Penny

Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies. WOW!

The US government needs to look for a leprechaun with a magic penny.

Parable 3

Bacteria in a Bottle

Parable 3 Suppose you place a single bacterium in a bottle at 11:00 am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on.

Question0: If the bottle is full at NOON, how many bacteria are in the bottle?

Question1: When was the bottle half full?

Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?

Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

Question0: If the bottle is full at NOON, how many bacteria are in the bottle?

Single bacteria in a bottle at 11:00 am

2 bacteria at 11:01

4 bacteria at 11:02

8 bacteria at 11:03

. . .

At 12:00 (60 minutes later) the bottle is full and contains260 ≈ 1.15 x1018

Question1: When was the bottle half full?

Single bacteria in a bottle at 11:00 am

2 bacteria at 11:01

4 bacteria at 11:02

8 bacteria at 11:03

. . .

Bottle is full at 12:00 (60 minutes later)

and so is 1/2 full at 11:59 am

Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?

½ full at 11:59

¼ full at 11:58

⅛ full at 11:57

full at 11:56

At 11:56 the amount of unused space is 15 times the amount of used space.

Your mathematically unsophisticated bacteria friends will not believe you when you warn of impending disaster at 11:56.

Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

There are . . .

enough bacteria to fill 1 bottle at 12:00

enough bacteria to fill 2 bottles at 12:01

enough bacteria to fill 4 bottles at 12:02

They have gained only 2 additional minutes for the bacteria civilization.

Question4: Is this scary?

By 1:00- there are 2120 bacteria.

This is enough bacteria to cover the entire surface of the Earth in a layermore than 2 meters deep!

After 5 ½ hours, at this rate . . .

the volume of bacteriawould exceed the volume of the known universe.

Yes, this is scary!

Key Facts about Exponential Growth

• Exponential growthcannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.

• Exponential growthleads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.

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