Exponential growth: the growth factor

Science and society, grade 11 (3 units) Lesson 1 of 16Learning

Exponential growth: the growth factor

Bacteria that double their number

A bacterium splits in two, and each of the two splits again. So in each fixed stretch of time the number of bacteria is multiplied.

A dish in a laboratory holds 500 bacteria.
Every hour the number of bacteria doubles.

Hours from the start	0	1	2	3	4
Number of bacteria	500	1,000	2,000	4,000	8,000

Each value in the table is the one before times 2.
After 4 hours we have multiplied by 2 four times: $\displaystyle {500\cdot2\cdot2\cdot2\cdot2}={500\cdot2^{4}}={8000}$ .
After 10 hours: $\displaystyle {500\cdot2^{10}}={500\cdot1024}={512{,}000}$ – more than 1,000 times the start.

An exponential process and the growth factor

An exponential process is one in which the amount is multiplied by the same number in each unit of time. That number is called the growth factor, written $\displaystyle q$ .

For the bacteria the unit of time is an hour, and the growth factor is $\displaystyle {q}={2}$ .
When the amount grows, $\displaystyle q$ is greater than 1. When it shrinks, $\displaystyle q$ is between 0 and 1 (in the lesson on decay).
In exponential growth the amount grows by more each hour: from 500 to 1,000 500 bacteria were added, and from 4,000 to 8,000, 4,000.

The formula of an exponential process

After $\displaystyle t$ units of time, the amount at the start has been multiplied by the growth factor $\displaystyle t$ times:

\displaystyle M_t=M_0\cdot q^{t}

$\displaystyle M_0$ – the amount at the start;
$\displaystyle q$ – the growth factor;
$\displaystyle M_t$ – the amount after $\displaystyle t$ units of time.

200 users signed up to a new social network in its first week.
Every week the number of users triples.
How many users will there be after 6 weeks?

M_6=200\cdot3^{6}=200\cdot729=145{,}800

On a calculator a power is worked out with the $\displaystyle x^y$ key (or $\displaystyle \wedge$ ): 3, then the key, then 6.

Recognising an exponential table

In the table of an exponential process, each value divided by the one before gives the same number – the growth factor.

The number of residents of a new town, at the start of each year:

Year	0	1	2	3
Residents	4,000	4,400	4,840	5,324

$\displaystyle {\frac{4400}{4000}}={1.1}$ , $\displaystyle {\frac{4840}{4400}}={1.1}$ , $\displaystyle {\frac{5324}{4840}}={1.1}$ – each year the number of residents is multiplied by 1.1.
The process is exponential, with $\displaystyle {M_0}={4000}$ and $\displaystyle {q}={1.1}$ .
The differences are not equal: 400 were added, then 440, then 484.

Exercise 1

An experiment in a laboratory starts with 300 bacteria.
Every hour the number of bacteria triples.

a

How many bacteria will there be after 2 hours?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many bacteria will there be after 5 hours?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The number of views of a video on the day it went online, and on the three days after:

Day	0	1	2	3
Views	800	1,200	1,800	2,700

a

Work out the growth factor: divide the number of views on day 1 by the number on day 0.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

If the growth goes on like this, how many views will there be on day 5?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A rumour spreads through a school.
On the first day two pupils know it.
Every day the number who know it triples.

How many pupils know the rumour after 6 days?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Here are two tables of an amount that changes with time.

Time	0	1	2	3
Amount A	50	100	150	200
Amount B	50	100	200	400

Which of the amounts changes in an exponential process? Explain by dividing each value by the one before.

Exercise 5

The number of bees in a hive is multiplied by 1.2 every month.
After one month there were 6,000 bees in the hive.

Write an equation and solve it: how many bees were there in the hive at the start?

Formulas: science and society, grade 11 (3 units)

Growth and decay

An exponential process – $\displaystyle M_0$ the amount at the start, $\displaystyle q$ the factor the amount is multiplied by in each unit of time, $\displaystyle M_t$ the amount after $\displaystyle t$ units of time
$\displaystyle M_t=M_0\cdot q^{t}$
The growth factor – a growth of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a decay of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1-\frac{p}{100}$
Decay by half-life – $\displaystyle n$ the number of half-lives gone by
$\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}$
The number of half-lives – $\displaystyle t$ the time gone by, $\displaystyle T$ the half-life
$\displaystyle n=\frac{t}{T}$
Linear growth – the amount grows by $\displaystyle d$ in each unit of time
$\displaystyle M_t=M_0+d\cdot t$

Measures of spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range – the greatest datum minus the least
$\displaystyle x_{\max}-x_{\min}$
The deviation of the datum $\displaystyle x_i$ from the mean
$\displaystyle x_i-\bar{x}$
The standard deviation of $\displaystyle n$ data
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}$
The standard deviation from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}$

Probability

The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities on its branches
$\displaystyle P=p_1\cdot p_2\cdot p_3$
“At least once” in $\displaystyle n$ independent experiments – $\displaystyle p$ the probability that it happens in each
$\displaystyle 1-(1-p)^{n}$
Conditional probability – the probability of $\displaystyle A$ when $\displaystyle B$ is known to have happened; $\displaystyle A\cap B$ – both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
The probability of “both $\displaystyle B$ and $\displaystyle A$ ”
$\displaystyle P(A\cap B)=P(B)\cdot P(A|B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)=P(A)$
The expected value – $\displaystyle x_1, \dots, x_k$ the possible values, $\displaystyle p_1, \dots, p_k$ their probabilities
$\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k$