# Continuous Exponential Decay: Time

How to find the time of continuous exponential decay: formula, 1 example, and its solution.

## Formula

### Formula

Just like finding the time of

continuous exponential growth,

to find the time of

continuous exponential decay,

use the exponential change formula.

A_{0}e^{rt} = A

A_{0}: Initial value

r: Rate of change (per time period)

t: Number of time period

A: Final value

## Example

### Example

### Solution

The initial value of the weight is 50g.

So A_{0} = 50g.

It says

after how many seconds will the weight

be less than 20g?

So set A = 20g.

The weight decreases

at a rate of 3% per second.

So r = -0.03/second.

The minus sign means decreasing.

A_{0} = 50

A = 20

r = -0.03

The weight decreases continuously.

Then 50⋅e^{-0.03⋅t} = 20.

The goal is to find the time t.

Divide both sides by 50.

20/50

= 2/5

= 4/10

= 0.4

e^{-0.03t} = 0.4

Then -0.03t = ln 0.4.

Logarithmic Form

Natural Logarithm

It says

assume ln 0.4 = -0.916.

Then -0.03t = -0.916.

Multiply -1 to both sides.

Divide both sides by 0.03.

Move the decimal points

2 digits to the right.

0.916/0.03 = 91.6/3

Find the value of 91.6/3

to the ones.

91.6/3 = 30.xx

t = 30.xx

Round this up to the nearest ones:

30.xx → 31.

The unit of the time is [second].

So write

After 31 seconds.

t = 30.xx means

after 30.xx seconds,

the weight will be exactly 20g.

So after 31 seconds,

the weight will be less than 20g.

So

after 31 seconds

is the answer.