# Exponential Growth and Decay

## Exponential Growth and Decay Revision

**Exponential Growth and Decay**

Exponential **growth** and **decay** describe the change of a current value.

Exponential **growth** is when the growth rate increases in proportion to the growing total or number.

Exponential **decay** is the opposite, where the decay rate decreases in proportion to the current total or number.

Make sure you are happy with the following topics before continuing:

**Exponential Growth**

When a value is **growing exponentially**, the greater the quantity, the faster the value grows. We often see **exponential growth** in population models.

**Example:**

A population of ducks is **growing exponentially**. Originally, there were \textcolor{blue}{6} ducks. The number of ducks doubles each year.

As the number of ducks doubles each year, the population of ducks will always be multiplied by a power of 2. We can model this **exponentially** with the equation:

\textcolor{purple}{D} = \textcolor{blue}{A}\times 2^\textcolor{orange}{x}

\textcolor{purple}{D} = number of ducks after \textcolor{orange}{x}\text{ years}

\textcolor{blue}{A} = initial number of ducks

For example, we could work out how many ducks there would be after 3 years by inputting \textcolor{orange}{x}=3

\textcolor{purple}{D} = \textcolor{blue}{6}\times 2^\textcolor{orange}{3}

\textcolor{purple}{D} = \textcolor{blue}{6}\times 2^\textcolor{orange}{3}

\textcolor{purple}{D} = 48

**Exponential Decay**

When a value is **decaying exponentially**, the lower the quantity, the slower the value decreases. We may also see **exponential decay** in population models for endangered species.

**Example:**

A population of elephants is **decaying exponentially**. There are currently \textcolor{blue}{72900} elephants. The number of elephants divides by 3 each year.

As the number of elephants divides by 3, the population size will be multiplied by a power of \dfrac{1}{3}

\textcolor{purple}{E} = \textcolor{blue}{A}\times \dfrac{1}{3}^\textcolor{orange}{x}

This may also be written as:

\textcolor{purple}{E} = \textcolor{blue}{A}\times 3^\textcolor{orange}{-x}

\textcolor{purple}{E} = number of elephants after \textcolor{orange}{x}\text{ years}

\textcolor{blue}{A} = initial number of elephants

For example, we could work out how many elephants there would be after 5 years by inputting \textcolor{orange}{x}=-5

\textcolor{purple}{E} = \textcolor{blue}{72900}\times 3^\textcolor{orange}{-5}

\textcolor{purple}{E} = \textcolor{blue}{72900}\times 3^\textcolor{orange}{-5}

\textcolor{purple}{E} =300

**Example 1: Exponential Growth**

A colony of bacteria increases exponentially each hour. The colony had 500 bacteria initially, and increased to 32,000 after 3 hours. Calculate by how much the population increases per hour.

**[3 marks]**

We can use the equation we found earlier:

\textcolor{purple}{32,000} = \textcolor{blue}{500}\times r^\textcolor{orange}{3}

Where r is the population increase per year.

64 = r^\textcolor{orange}{3}

Cube root each side:

\sqrt[3]{64} = r\\

r = 4

**Example 2: Exponential Decay**

Radioactive substances become less toxic over time, and the time taken for them to become half as toxic is called their half-life. Carbon has a half life of 5730\text{ years}. If a sample contains 1\text{ g} of carbon, work out how many grams of carbon it would have after 22920\text{ years}.

**[2 marks]**

The unit of time used in this exponential decay is 5730\text{ years}, so we can work out how many of these units we are working out the exponential decay for:

22920\div 5730 =4 units of time.

We can write this as an equation:

\textcolor{purple}{A} = \textcolor{blue}{1}\times \dfrac{1}{2}^\textcolor{orange}{4}

\textcolor{purple}{A} = \textcolor{blue}{1}\times 2^\textcolor{orange}{-4}

Where A is the amount of Carbon.

\textcolor{purple}{A} = 2^\textcolor{orange}{-4}\\

\textcolor{purple}{A} = \dfrac{1}{16}\text{ g}

## Exponential Growth and Decay Example Questions

**Question 1:** A population of bacteria grows exponentially by 4 times each hour. Between which two hours does the population size increase the most?

(a) Between hours 1 and 2

(b) Between hours 4 and 5

(c) Between hours 6 and 7

(d) Between hours 8 and 9

**[1 marks]**

We know with exponential growth, the greater the quantity, the faster it grows, so the biggest difference will be between 8 and 9 (d).

This means, the population grows most between 8 and 9

**Question 2:** A population of bears increases exponentially by tripling each year. After 6 years, the population had 583200 bears. Calculate how many bears there were initially.

**[3 marks]**

We can form an equation:

583200=A\times 3^6\\ A=\dfrac{583200}{3^6}\\A=800 bears initially.

**Question 3:** A population of killer whales is exponentially decaying by halving each year. When there are less than 3000 killer whales, the population is considered critically endangered. The population is currently 100,000 killer whales.

Will the population be critically endangered in 5\text{ years}?

**[2 marks]**

We can form an equation:

W = 100,000\times 2^{-5}\\ W =3125Therefore, as the population would be higher than 3000, the killer whales will not be critically endangered.