How to Solve Exponential Growth and Decay Functions?

A step-by-step guide to exponential growth and decay

The formulas of exponential growth and decay are presented below:

• Exponential growth:\(\color{blue}{f(x)=a(1+r)^t}\)
• Exponential decay:\(\color{blue}{f(x)=a(1- r)^t}\)

Exponential growth uses a factor \(r\) which is the rate of growth. The \(r\)-value lies between \(0\) and \(1\) \((0<r<1)\). The expression \((r + 1)\) can be considered a growth factor. And \(t\) is the time step which is the number of times the growth factor is to be multiplied. The value of \(t\) can be a whole number or a decimal number. For exponential decay, the growth factor is \((1 – r)\), which has a value less than \(1\).

Exponential Growth and Decay – Example 1:

If \($100,000\) is invested at a compound rate of \(4%\) per quarter, after \(2\) years, what is the amount received from the investment fund?

Solution:

The invested principal is \(a=$100,000\), the rate of compounding growth is \(r= 4%= 0.04\) per quarter.

The time is \(2\) years, and there are \(4\) quarters in a year, and we have \(t= 8\).

Using the concepts of exponential growth and decay, we have the following expressions for exponential growth:

\(f(x)=a(1 + r)^t\)

\(f(x)=100,000(1 + 0.04)^8\)

\(=100,000 (1.04)^8\)

\(=136856.90504\)

Therefore an amount of \($136,857\) is received after a period of \(2\) years.

Exercises for Exponential Growth and Decay

1. The radioactive substance of thorium decomposes at a rate of \(7%\) per minute. What portion of \(15\) grams of thorium remains after \(6\) minutes?
2. In \(2020\), \(200\) people lived in a remote town. The population has increased by \(20%\) every year. How many people will live in \(5\) years?

1. \(\color{blue}{9.7}\)
2. \(\color{blue}{498}\)