# Exponential Decay (Part 1)

How to solve exponential decay problems: formulas, examples, and their solutions.

## Formula

*A* = *A*_{0}(1 + *r*)^{t}*A*: finla value*A*_{0}: initial value*r*: rate of change (per period)*t*: number of period

The difference between

the exponential growth and decay

is the sign in front of *r*:

+*r* → growth

-*r* → decay

## Proof

Period 1

(End) = *A*_{0} - *A*_{0}⋅*r*

= *A*_{0}(1 - *r*)^{1}

Period 2

(End) = *A*_{0}(1 - *r*) - *A*_{0}(1 - *r*)⋅*r*

= *A*_{0}(1 - *r*)(1 - *r*)

= *A*_{0}(1 - *r*)^{2}

Period 3

(End) = *A*_{0}(1 - *r*)^{2} - *A*_{0}(1 - *r*)^{2}⋅*r*

= *A*_{0}(1 - *r*)^{2}(1 - *r*)

= *A*_{0}(1 - *r*)^{3}

See the pattern

between the 'Period' and the 'End'.

Period *t*

(End) = *A*_{0}(1 - *r*)^{t}

Inductive reasoning

## Example 1

Write the given values to use.

Write the period unit 'min'

when writing *r* and *t*.

Put the given values into the formula.*A* = 80⋅(1 - 0.05)^{60}

Use 0.95^{60} = 0.046.

(given condition)

Write the answer with its unit: 3.68 g.

## Formula: Continuous Decay

For continuous decay,

use the continuous exponential decay formula:*A* = *A*_{0}*e*^{-rt}.

(Difference: *rt* → -*rt*)

Exponential growth (part 1)

Definition of *e* (mathematical constant)

## Proof: Continuous Decay

Start from the formula with *n*.

Change the exponent:*t*⋅*n* → (-*n*/*r*)⋅(-*r*⋅*t*).

Then, as *n* → ∞,

(1 - *r*/*n*)^{-n/r} (dark gray) becomes *e*.*e* = 2.718...

So *A* = *A*_{0}*e*^{-rt}.

## Example 2

Write the given values to use.

The term 'continuously' is used.

So use the continuous compounding formula.*A* = 80⋅*e*^{-0.05⋅60}

Use *e*^{-3} = 0.05.

(given condition)

Write the answer with its unit: 4.0 g.

As you can see,

the answer is greater than

the previous example's answer:

the amount of decay is minimized by**continuous** decaying.