6.4 Exponential Growth and Decay

Greg Kelly, Hanford High School, Richland, Washington

Glacier National Park, MontanaPhoto by Vickie Kelly, 2004

The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep present (at least for awhile.)

So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account.

If the rate of change is proportional to the amount present, the change can be modeled by:

dy kydt

dy kydt

1 dy k dty

1 dy k dty

ln y kt C

Rate of change is proportional to the amount present.

Divide both sides by y.

Integrate both sides.

1 dy k dty

ln y kt C

Integrate both sides.

Exponentiate both sides.

When multiplying like bases, add exponents. So added exponents can be written as multiplication.

ln y kt Ce e

C kty e e

ln y kt Ce e

C kty e e

Exponentiate both sides.

When multiplying like bases, add exponents. So added exponents can be written as multiplication.

C kty e e

kty Ae Since is a constant, let .Ce Ce A

C kty e e

kty Ae Since is a constant, let .Ce Ce A

At , .0t 0y y00

ky Ae

0y A

1

0kty y e This is the solution to our original initial

value problem.

0kty y eExponential Change:

If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.

Note: This lecture will talk about exponential change formulas and where they come from. The problems in this section of the book mostly involve using those formulas. There are good examples in the book, which I will not repeat here.

Continuously Compounded Interest

If money is invested in a fixed-interest account where the interest is added to the account k times per year, the amount present after t years is:

0 1ktrA t A

k

If the money is added back more frequently, you will make a little more money.

The best you can do is if the interest is added continuously.

Of course, the bank does not employ some clerk to continuously calculate your interest with an adding machine.

We could calculate: 0lim 1kt

k

rAk

but we won’t learn how to find this limit until chapter 8.

Since the interest is proportional to the amount present, the equation becomes:

Continuously Compounded Interest:

0rtA A e

You may also use:rtA Pe

which is the same thing.

(The TI-89 can do it now if you would like to try it.)

Radioactive Decay

The equation for the amount of a radioactive element left after time t is:

0kty y e

This allows the decay constant, k, to be positive.

The half-life is the time required for half the material to decay.

Half-life

0 012

kty y e

1ln ln2

kte

ln1 ln 2 kt 0

ln 2 kt

ln 2 tk

Half-life:

ln 2half-lifek

Newton’s Law of Cooling

Espresso left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air.

(It is assumed that the air temperature is constant.)

If we solve the differential equation: sdT k T Tdt

we get:Newton’s Law of Cooling

0kt

s sT T T T e

where is the temperature of the surrounding medium, which is a constant.

sT