# Decay Formula

## Decay Formula

Students should first go over the definition of exponential decay before learning the exponential Decay Formula. In exponential decay, a quantity initially drops gradually before decreasing quickly. In addition to using the exponential Decay Formula to calculate population decline (depreciation), students can also use it to calculate half-life (the amount of time for the population to become half of its size).

The exponential decline, which is a rapid reduction over time, can be calculated with the use of the exponential Decay Formula. The exponential Decay Formula is used to determine population decay, half-life, radioactivity decay, and other phenomena. f(x) = a (1 – r)x is the general form.

Where

a = initial amount

1-r = decay factor

x= time period

### Decay Formula

A variable exponent, a positive base, and a base that is not equal to one are all characteristics of an exponential function. Since the exponent in F (x) = 4x is fixed rather than changeable, it is an example of an exponential function. f (x) = x3 is a basic polynomial function rather than an exponential function. Exponential function graphs never reach a horizontal asymptote and are continually curved. Exponential or logarithmic functions can be used to explain a variety of real-world occurrences.

In Mathematics, the concept of “exponential decay” describes the process of a constant percentage rate decline in an amount over time. It can be written as y=a(1-b)x, where x is the amount of time that has passed, and is the initial amount, b is the decay factor, and y is the final amount.

The exponential Decay Formula is helpful in many real-world situations, most notably for keeping track of inventory that is consumed consistently in the same amount (such as food for a school cafeteria). It is also particularly helpful in that it can be used to quickly calculate the overall cost of using a product over time.

What are Exponential Decay Formulas?

The quantity drops gradually, followed by a dramatic decline in the rate of change and growth over time. The exponential Decay Formula is used to determine this decline in growth. One of the following formats can be used to express the exponential Decay Formula:

f(x) = abx

f(x) = a (1 – r)x

P = P

0

e-k t

Where,

a (or) P

0

= Initial amount

b = decay factor

r = Rate of decay (for exponential decay)

x (or) t = time intervals (time can be in years, days, (or) months, whatever students are using should be consistent throughout the problem).

k = constant of proportionality

e- Euler’s constant

### Exponential Decay Formula

Understanding the Exponential Decay Formula and being able to name each of its components is crucial:

y = a (1-b)x

The term “decay factor” (represented by the letter b in the exponential Decay Formula), which is a percentage by which the initial amount will decline each time, is crucial to comprehend in order to fully appreciate the utility of the Decay Formula.

If students are thinking about this practice, the original amount would be the number of apples a bakery purchases, and the exponential factor would be the proportion of apples used each hour to make pies. The original amount, denoted by the letter and in the formula, is the amount before the decay occurs.

The exponent indicates how frequently the decay occurs and is typically stated in seconds, minutes, hours, days, or years. In the case of exponential decay, the exponent is always time and is represented by the letter x.

In exponential decay, a quantity first decreases gradually before quickly doing so. The exponential Decay Formula can be used to calculate half-life as well as population decline (depreciation) (the amount of time for the population to become half of its size).

### Examples Using Exponential Decay Formulas

• The half-life of carbon-14 is 5,730 years. Find the exponential decay model for carbon-14. To the nearest decimal point, please round the response.

Solution: Use the formula of exponential decay

P = P0 e– k t

P0 = initial amount of carbon

The half-life of carbon-14 is 5,730 years,

P = P0 / 2 = Half of the initial amount of carbon when t = 5, 730.

P0 / 2 =P0 e– k (5730)

Divide both sides by P0

0.5 = e– k (5730)

Take “ln” on both sides,

ln 0.5 = -5730k

Divide both sides by -5730,

k = ln 0.5 / (-5730) ≈ 1.2097

The exponential decay model of carbon-14 is P = P0 e– 1.2097k

• A new couch cost $350,000 from Andrew. The sofa’s value is decreasing exponentially at a rate of 5% yearly. What is the sofa worth now, two years later? To the nearest decimal point, please round the response. Solution: Initial value of Sofa=$350,000

Rate of decay r = 5% = 0.05

Time t = 2 years

Use the exponential Decay Formula,

A = P (1 – r)t

A = 350000 x (1 – 0.05)

2

A = 315,875

The value of the sofa after 2 years = \$315,875