Growth and decay

In a process of growth or decay the quantity is multiplied by the same number $\displaystyle q$ in every unit of time – every year, every day or every hour. The number $\displaystyle q$ is called the growth factor (when $\displaystyle q>1$) or the decay factor (when $\displaystyle 0<q<1$).

• A quantity that grows by $\displaystyle p$ per cent in every unit of time is multiplied by $\displaystyle {q}={1+\frac{p}{100}}$: growth of 5% – $\displaystyle {q}={1.05}$.
• A quantity that shrinks by $\displaystyle p$ per cent is multiplied by $\displaystyle {q}={1-\frac{p}{100}}$: a fall of 20% – $\displaystyle {q}={0.8}$.

After $\displaystyle t$ units of time the quantity has been multiplied $\displaystyle t$ times by $\displaystyle q$:

• $\displaystyle f(0)$ – the quantity at the start.
• $\displaystyle f(t)$ – the quantity after $\displaystyle t$ units of time; $\displaystyle t$ may be a fraction, and negative too – a time before the measuring began.

A bacterium splits in two, and each part becomes a bacterium of its own. Suppose the laboratory has one bacterium, and the population doubles every hour:

• After $\displaystyle t$ hours there are $\displaystyle 2^t$ bacteria: 1, 2, 4, 8, 16, … and after a day $\displaystyle {2^{24}}={16{,}777{,}216}$.
• The numbers at the whole hours are a geometric sequence with the ratio $\displaystyle {q}={2}$: equal steps in $\displaystyle t$ multiply the quantity by the same number.

When there are $\displaystyle 100{,}000$ bacteria at the start, $\displaystyle {f(t)}={100000\cdot2^t}$:

• An hour before the experiment began: $\displaystyle {f(-1)}={100000\cdot2^{-1}}={50{,}000}$ bacteria.
• In half an hour the population grows $\displaystyle 2^{0.5}\approx1.414$ times, and in a quarter of an hour $\displaystyle 2^{0.25}\approx1.189$ times.

In a culture of bacteria 400 bacteria were measured.
3 hours later 1,350 bacteria were measured in the same culture.
The number of bacteria grows by a fixed percentage every hour.

The example below finds the growth factor per hour.

• $\displaystyle {400\cdot q^3}={1350}$, so $\displaystyle {q^3}={\frac{1350}{400}}={3.375}$.
• $\displaystyle {q}={3.375^{\frac{1}{3}}}={1.5}$: the number of bacteria grows by 50% every hour.
• The function: $\displaystyle {f(t)}={400\cdot1.5^t}$, where $\displaystyle t$ is the hours since the first measurement.

The doubling time is the time in which a growing quantity doubles. The half-life $\displaystyle T$ is the time in which a decaying quantity shrinks to half:

A patient’s body holds 100 mg of a medicine, and every hour the amount shrinks by 20%. The example below finds the medicine’s half-life.

• $\displaystyle {0.8^T}={0.5}$, that is $\displaystyle {T}={\log_{0.8}0.5}$.
• On a calculator, changing to base 10: $\displaystyle {T}={\frac{\log0.5}{\log0.8}\approx3.106}$ hours.

• The half-life does not depend on the starting amount: every 3.106 hours the amount halves – from 100 to 50, from 50 to 25.
• When the half-life is given, $\displaystyle q$ follows from it. Carbon-14 has a half-life of 5,730 years: $\displaystyle {q^{5730}}={0.5}$, so $\displaystyle {q}={0.5^{\frac{1}{5730}}\approx0.999879}$ a year.

In $\displaystyle {f(t)}={f(0)\cdot q^t}$ the time $\displaystyle t$ is measured in the unit in which $\displaystyle q$ is given.

• Interest of 5% a year: after 18 months $\displaystyle {t}={1.5}$ years have passed, and 12,000 ₪ have grown to $\displaystyle 12000\cdot1.05^{1.5}\approx12911.16$ ₪.
• The growth factor per month is the number that, multiplied by itself 12 times, gives $\displaystyle 1.05$: $\displaystyle 1.05^{\frac{1}{12}}\approx1.00407$.
• So the sum grows by about 0.41% a month, not by $\displaystyle \frac{5}{12}\%$.

A town has 80,000 inhabitants.
The number of inhabitants grows by 2.5% every year.

How many inhabitants will the town have in 8 years? Write the calculation, and round to a whole number.

In how many years will the town have 100,000 inhabitants? Solve the equation, and round the answer to two decimal places.

At the start of an experiment a dish held 500 bacteria.
After 4 hours it held 2,000 bacteria.
The number of bacteria grows by a fixed percentage every hour.

Calculate the growth factor per hour.

How many bacteria will the dish hold after 10 hours?

$\displaystyle {\left(\sqrt{2}\right)^t}={2^{\frac{t}{2}}}$.
Find the doubling time: solve the equation.

A patient took sleeping pills, and his blood holds 4 milligrams of the medicine.
Every hour the amount of the medicine in the blood shrinks by 25%.

How many milligrams of the medicine will be in the blood after 6 hours? Round to three decimal places.

Find the half-life of the medicine.

A blood test finds the medicine when the blood holds at least 0.1 milligrams of it.
Solve the inequality: when will the test come out negative?

A savings plan pays interest of 6% a year.
10,000 ₪ were deposited in the plan.

What is the growth factor per month? Round to five decimal places.

How much money will the plan hold after 30 months? Round to the agora.