Exponential Decay (Part 1)

Exponential Decay (Part 1)

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

Formula

A = A_0(1 - r)^t. A: Final Value, A_0: Initial Value, r: Rate of Change (per Period), t: Number of Period

A = A0(1 + r)t

A: finla value
A0: 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

Exponential Decay: Proof of the Formula

Period 1

(End) = A0 - A0r
= A0(1 - r)1

Period 2

(End) = A0(1 - r) - A0(1 - r)⋅r
= A0(1 - r)(1 - r)
= A0(1 - r)2

Period 3

(End) = A0(1 - r)2 - A0(1 - r)2r
= A0(1 - r)2(1 - r)
= A0(1 - r)3

See the pattern
between the 'Period' and the 'End'.

Period t

(End) = A0(1 - r)t

Inductive reasoning

Example 1

A radioactive substance weighs 80g. If it is decreasing at a rate of 5% per minute, find the expected weight 1 hour later. (Assume 0.95^60 = 0.047.)

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 0.9560 = 0.046.
(given condition)

Write the answer with its unit: 3.68 g.

Formula: Continuous Decay

A = A_0*e^rt. A: Final Value, A_0: Initial Value, r: Rate of Change (per Period), t: Number of Period

For continuous decay,
use the continuous exponential decay formula:

A = A0e-rt.
(Difference: rt → -rt)

Exponential growth (part 1)

Definition of e (mathematical constant)

Proof: Continuous Decay

Exponential Decay (Part 1): Proof of the Formula (Continuous Decay)

Start from the formula with n.

Change the exponent:
tn → (-n/r)⋅(-rt).

Then, as n → ∞,
(1 - r/n)-n/r (dark gray) becomes e.

e = 2.718...

So A = A0e-rt.

Example 2

A radioactive substance weighs 80g. If it is continuously decreasing at a rate of 5% per minute, find the expected weight 1 hour later. (Assume e^-3 = 0.05.)

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.