# Section 9-6

# Exponential Growth and Decay

## EXPONENTIAL GROWTH

Exponential growth models are often used for real-world situations like interest earned on an investment, human or animal population, bacterial culture growth, etc.

The general exponential growth model is

*y* = *C*(1 + *r*)^{t},

where *C *is the initial amount or number, *r* is the growth rate (for example, a 2% growth rate means *r* = 0.02), and *t* is the time elapsed.

Example:

A population of 32,000 with a 5% annual growth rate would be modeled by the equation:

*y* = 32000(1.05)^{t}

with *t* in years.

Sometimes, you may be given a doubling or tripling rate rather than a growth rate in percent. For example, if you are told that the number of cells in a bacterial culture doubles every hour, then the equation to model the situation would be:

*y* = *C* · 2^{t}

with *t* in hours.

## EXPONENTIAL DECAY

Exponential decay models are also used very commonly, especially for radioactive decay, drug concentration in the bloodstream, of depreciation of value.

## Radioactive Decay

Radioactive decay problems are often given in terms of half-life of a radioactive element. This is modeled by the equation:

** **

where *N*_{0} is the initial amount of the element, *N* is the amount remaining after *t* years, and τ is the half-life.

Example:

If you start with a quantity of the unstable element Potassium-40, it takes 1.26 billion years for half of it to decay into Argon-40. So the **half-life** of Potassium-40 is 1.26 billion years.

Write an exponential decay model to find the number of Potassium-40 atoms remaining after

*t *years, if you start with 2000 Potassium-40 atoms.

Here, *N*_{0} = 2000 and τ = 1,260,000,000. So the model is:

## Drug Concentration

For drug concentration problems, you may be given the fraction *p* of the original amount of the drug left in the bloodstream after a unit of time. In this case, the situation is modeled by the equation

*y *= *A**p*^{t},

where *y* is the concentration remaining after time *t*, and *A* is the initial amount*.*

Example:

If a person takes *A *milligrams of a drug at time 0, then *y *= *A*(0.8)^{t} gives the concentration left in the bloodstream after *t* hours. If the initial dose is 200 mg, what is the concentration of the drug in the bloodstream after 4 hours?

Substitute.

**y**** = 200(0.8)**^{4}

You might want a calculator!

**y**** = 200(0.4096)**

**y**** = 81.92**

So there are about 82 milligrams of the drug left in the bloodstream after four hours.

## Depreciation

If the value of some article (for example, a car), originally $*C*, depreciates *x*% per year, then the value after *t* years is given by the formula:

*y* = *C*(1 –** ***x*/100)^{t}

Example:

The original value of a car is $28,000. If it depreciates by 15% each year, find its value in 4 years.

Substitute.

**y**** = 28000(1 **–** 0.15)**^{4}

**y**** = 28000(0.85)**^{4}

**y**** = 28000(0.52200625) **

**y**** = 14616.175 **

So after four years, the car is worth about $14,616.