Decay in percent

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

Decay in percent

The decay factor

When an amount shrinks by $\displaystyle p$ percent, $\displaystyle 100\%-p\%$ of it is left. So a decay of $\displaystyle p\%$ is multiplying by the decay factor, a number between 0 and 1:

\displaystyle q=1-\frac{p}{100}

A patient was given a medicine, and there are now 400 mg of it in his blood.
Every hour the liver breaks down 20% of the medicine in the blood.

The decay factor: $\displaystyle {q}={1-\frac{20}{100}}={0.8}$ – every hour 80% of what there was is left.

Hours	0	1	2	3	4
Medicine in the blood (mg)	400	320	256	204.8	163.84

After 5 hours: $\displaystyle {400\cdot0.8^{5}}={131.072\approx131}$ mg.
Each hour the amount drops by less: in the first hour 80 mg went, and in the fourth about 41.

An amount that never reaches 0

In exponential decay part of what there was is left each hour, so the amount keeps shrinking but never reaches 0.

After 20 hours: $\displaystyle 400\cdot0.8^{20}\approx4.6$ mg – little, but not 0.
In practice, once the amount is very small it can no longer be measured, and doctors say the medicine has “left the body”.

A cup of tea cooling

A cup of tea cools down to the temperature of the room. What decays is the difference between the temperature of the tea and that of the room.

The room is at 20 degrees.
The tea is poured at 90 degrees – 70 degrees above the room.
Every minute the difference shrinks by 10%.

The difference after $\displaystyle t$ minutes: $\displaystyle 70\cdot0.9^{t}$ .
The temperature of the tea: the room plus the difference, $\displaystyle 20+70\cdot0.9^{t}$ .
After 5 minutes: $\displaystyle {20+70\cdot0.9^{5}}={61.3343\approx61.3}$ degrees.

The difference approaches 0, so the tea approaches 20 degrees – and never cools below the temperature of the room.

The percentage decay from a table

In a table of decay, a value divided by the one before gives the decay factor. The percentage decay is what $\displaystyle q$ lacks to reach 1, in percent.

Hours	0	1	2
Substance (g)	1,000	850	722.5

$\displaystyle {\frac{850}{1000}}={0.85}$ and $\displaystyle {\frac{722.5}{850}}={0.85}$ , so $\displaystyle {q}={0.85}$ .
$\displaystyle {0.85}={1-\frac{15}{100}}$ : every hour the amount shrinks by 15%.

Exercise 1

A patient’s blood holds 500 mg of a medicine.
Every hour the amount in the blood shrinks by 30%.

a

What is the decay factor?

/ – 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 mg of the medicine will be in the blood after 3 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

c

Will there be a time when the blood holds exactly 0 mg of the medicine? Explain.

Exercise 2

Because of pollution, the number of fish in a lake shrinks by 5% every year.
The lake now holds 8,000 fish.

How many fish will the lake hold after 6 years, if the pollution goes on? Round to a whole number.

/ – 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 bowl of hot soup was put in a room at 20 degrees.
At first the soup was 60 degrees hotter than the room.
Every minute the difference between the soup and the room shrinks by 20%.

What is the temperature of the soup after 4 minutes? Round to one decimal place.

/ – 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

The amount of a pollutant in a stream’s water, after sewage stopped flowing into it:

Weeks	0	1	2
Pollutant (kg)	2,000	1,700	1,445

a

What is the decay factor?

/ – 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

By what percentage does the amount of pollutant shrink every week?

/ – 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 5

Every hour the amount of a medicine in the blood shrinks by 20%.
An hour after the medicine was given, 240 mg were left in the blood.

Write an equation and solve it: how many mg of the medicine were given?

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
$\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k$