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## Exponential Function and Decay

In mathematics, exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula *y=a(1-b) ^{x }*wherein

*y*is the final amount,

*a*is the original amount,

*b*is the decay factor, and

*x*is the amount of time that has passed.

The exponential decay formula is useful in a variety of real world applications, most notably for tracking inventory that’s used regularly in the same quantity (like food for a school cafeteria) and it is especially useful in its ability to quickly assess the long-term cost of use of a product over time.

Exponential decay is different from linear decay in that the decay factor relies on a percentage of the original amount, which means the actual number the original amount might be reduced by will change over time whereas a linear function decreases the original number by the same amount every time.

It is also the opposite of exponential growth, which typically occurs in the stock markets wherein a company’s worth will grow exponentially over time before reaching a plateau. You can compare and contrast the differences between exponential growth and decay, but it’s pretty straightforward: one increases the original amount and the other decreases it.

## Elements of an Exponential Decay Formula

To start, it’s important to recognize the exponential decay formula and be able to identify each of its elements:

y = a (1-b)^{x}

In order to properly understand the utility of the decay formula, it is important to understand how each of the factors is defined, beginning with the phrase “decay factor”—represented by the letter *b* in the exponential decay formula—which is a percentage by which the original amount will decline each time.

The original amount here—represented by the letter *a *in the formula—is the amount before the decay occurs, so if you’re thinking about this in a practical sense, the original amount would be the amount of apples a bakery buys and the exponential factor would be the percentage of apples used each hour to make pies.

The exponent, which in the case of exponential decay is always time and expressed by the letter x, represents how often the decay occurs and is usually expressed in seconds, minutes, hours, days, or years.

## An Example of Exponential Decay

Use the following example to help understand the concept of exponential decay in a real-world scenario:

On Monday, Ledwith’s Cafeteria serves 5,000 customers, but on Tuesday morning, the local news reports that the restaurant fails health inspection and has—yikes!—violations related to pest control. Tuesday, the cafeteria serves 2,500 customers. Wednesday, the cafeteria serves only 1,250 customers. Thursday, the cafeteria serves a measly 625 customers.

As you can see, the number of customers declined by 50 percent every day. This type of decline differs from a linear function. In a linear function, the number of customers would decline by the same amount every day. The original amount (*a*) would be 5,000, the decay factor (*b* ) would, therefore, be .5 (50 percent written as a decimal), and the value of time (*x*) would be determined by how many days Ledwith wants to predict the results for.

If Ledwith were to ask about how many customers he would lose in five days if the trend continued, his accountant could find the solution by plugging all of the above numbers into the exponential decay formula to get the following:

*y = 5000(1-.5) ^{5}*

The solution comes out to 312 and a half, but since you can’t have a half customer, the accountant would round the number up to 313 and be able to say that in five days, Ledwith could expect to lose another 313 customers!

## Exponential Decay Formula

Before knowing the exponential decay formula, first, let us recall what is meant by an exponential decay. In exponential decay, a quantity slowly decreases in the beginning and then decreases rapidly. We use the exponential decay formula to find population decay (depreciation) and we can also use the exponential decay formula to find half-life (the amount of time for the population to become half of its size). Let us learn more about the exponential decay formula along with the solved examples

## What are Exponential Decay Formulas?

The Exponential decay formula helps in finding the rapid decrease over a period of time i.e. the exponential decrease. The exponential decay formula is used to find the population decay, half-life, radioactivity decay, etc. The general form is f(x) = a (1 – r)^{x}.

Where

a = initial amount

1-r = decay factor

x= time period

### Exponential Decay Formula

The quantity decreases slowly after which the rate of change and the rate of growth decreases over a period of time rapidly. This decrease in growth is calculated by using the exponential decay formula. The exponential decay formula can be in one of the following forms:

f(x) = ab^{x}

f(x) = a (1 – r)^{x}

P = P00 e^{–}^{k t}

Where,

- a (or) P00 = 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 you are using should be consistent throughout the problem).
- k = constant of proportionality
- e- Euler’s constant

Note: In exponential decay, always 0 < b < 1.Here, b = 1 – r ≈ e^{– k}.

## Examples Using Exponential Decay Formulas

**Example 1: Chris bought a new car for $20,000. The value of the car decreases exponentially (depreciates) at a rate of 8% per year. Then what is the value of the car after 5 years? Solve this by using exponential formulas and round your answer to the nearest two decimals.**

**Solution:**

The initial value of the car is, P = $20,000.

The rate of decay is, r = 8% = 0.08.

The time is t = 5 years.

Using the exponential decay formula:

A = P (1 – r)^{t}

A = 20000 (1 – 0.08)^{5 }= 13181.63

Therefore, the value of the car after 5 years = $13,181.63.

**Example 2: Jane bought a new house for $350,000. The value of the house decreases exponentially (depreciates) at a rate of 5% per year. Then what is the value of the house after 2 years? Solve this by using exponential formulas and round your answer to the nearest two decimals.**

**Solution:**

The initial value of the house = $3,50,000

The rate of decay, r = 5% = 0.05

The time, t = 2 years

Using the exponential decay formula,

A = P (1 – r)^{t}

A = 350000 (1 – 0.05)^{2}

A = 315,875

Therefore, the value of the house after 2 years = $315,875

**Example 3: The half-life of carbon-14 is 5,730 years. Find the exponential decay model of carbon-14. Solve it by using the exponential decay formula and round the proportionality constant to 4 decimals.**

**Solution:**

Using the given data, we can say that carbon-14 is decaying and hence we use the formula of exponential decay.

P = P00 e^{– k t}

Here, P00 = initial amount of carbon

It is given that the half-life of carbon-14 is 5,730 years. It means

P = P00 / 2 = Half of the initial amount of carbon when t = 5, 730.

Substitute all these values in the formula of exponential decay:

P00 / 2 =P00 e^{– k (5730)}

Dividing both sides by P00,

0.5 = e^{– k (5730)}

Taking “ln” on both sides,

ln 0.5 = -5730k

Dividing both sides by -5730,

k = ln 0.5 / (-5730) ≈ 1.2097

Thus, the exponential decay model of carbon-14 is P = P00 e^{– 1.2097k }

## FAQs on Exponential Decay Formulas

### What is Exponential Decay Formula?

The quantity decreases slowly at regular intervals by a fixed percent. This decrease in growth is calculated by using the exponential decay formula. The general form is f(x) = a (1 – r)^{x},

where

- a = Initial amount
- r = Rate of decay
- x = time interval

The exponential decay formula is used to find the population decay, half-life, radioactive decay, etc.

### What is the Formula For Finding the Exponential Decay?

The exponential decay formula helps in finding the rapid decrease over a period of time i.e. the exponential decrease. This decrease in growth is calculated by using the exponential decay formula. The exponential decay formula can be in one of the following forms:

f(x) = a (1 – r)^{x }(general form)

P = P00 e^{–}^{k t} (for continuous exponential decay)

where,

- a (or) P00 = Initial amount
- b = decay factor
- r = Rate of decay
- x (or) t = time intervals
- k = constant of proportionality
- e- Euler’s constant

### How Do You Calculate The Exponential Decay?

The exponential formula is y = ab^{x}. Here b is the decay factor. The decay is calculated as (1-r), where r = decay rate. Now y is the decay function. y = a (1-r)^{x}.

### How Do You Find The Decay Rate of An Exponential Function?

The exponential decay formula is f(x) = a b^{x }, where b is the decay factor. The decay rate in the exponential decay function is expressed as a decimal. The decay rate is given in percentage. We convert it into a decimal by just dropping off % and dividing it by 100. Then find the decay factor b = 1-r. For example, if the decay rate is 12%, then decay rate of the exponential function is 0.12 and the decay factor b= 1- 0.12 = 0.88

**Exponential Growth and Decay**

## Exponential Growth and Decay

## Growth and Decay

But sometimes things **can** grow (or the opposite: decay) exponentially, ** at least for a while**.

So we have a generally useful formula:

## Exponential Decay

Some things “decay” (get smaller) exponentially.

## Half Life

The “half life” is how long it takes for a value to halve with exponential decay.

Commonly used with radioactive decay, but it has many other applications!

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