Calculating Exponential Decay with a Variable In the Exponent

Exponents are used to represent any function that changes rapidly. If the process is for growth, it means that the bigger the process grows, the faster the growth. Exponential functions are explained with this formula:

y = abx

where

• y is the remaining quantity at the end of time t, say 2 years.
• a is the initial amount before the exponential was applied. If it is for decay, then a is the initial quantity before the decay started.
• b is the rate of decay or growth.
• x is the number of time intervals

Exponential Decay

The example given above is a general pattern for an exponential function. Exponential decay occurs when an initial quantity is reduced at a constant rate (in percentage) over a period.

Imagine that you have a bag full of chocolate. Each day you go to school, you take half of the chocolate in the bag to school. So on the first day, you took half of the chocolate in the bag to school, and on the second day, you took half of the chocolate that remained from yesterday.

If you continue taking half each day, you will soon have less than a bar of chocolate in your bag. That is a good example of exponential decay. An exponentially decaying function should be written like this:

• y = a ( 1 – r )x
• Note that b = ( 1 – r )

Where r is the rate of decay in percentage. The percentage is usually converted to a decimal fraction. Let’s look at the proper calculations of exponential decays. We'll start with decays that are exponential but not continuous.

Non-Continuous Exponential Decay

In a non-continuous exponential decay, the decay takes place at the end of a specific time, such as an hour, day, or month. Like in the chocolate example given above, the number of chocolates in the bag changes at the end of the day. So the time of change is 1 day. It's a non-continuous decay.

Example 1

Imagine you were born in a large town where you could not easily count the number of people in the town. From records, the total number of people living in your town was 180,000 when you were born.

Then, due to climate change, there was increased flooding and hurricanes. So people have been gradually leaving the town, and the population decline rate has been 2% at the end of every year since you were born.

What is the current number of people in your town if you are 9 years old?

Step 1

The first step is to write your formula for exponential decay.

y = a ( 1 – r )x

Let us get the data required for the solution from the question. We are looking for y which is the number of people remaining after 9 years.

• a = 180,000
• which is the initial number of people that were in your town before the decay started.
• r = 2%
• = 0.02
• The rate of decay must be converted to a decimal fraction.
• x = 9

The number of time intervals is 9, even though a year consists of 12 months. We're using years because the population changes by 2% only at the end of the year. So the period is 1 year. If the population was changing at the end of every month, then we would have counted the number of months.

Step 2

The next step is to input the values for the variables in the formula and calculate

• y = a ( 1 – r )x
• y = 180,000 * ( 1 – 0.02 )9
• y = 180,000 * 0.989
• y = 180,000 * 0.834
• y = 150,074

So at the end of 9 years, the number of people remaining in your town is 150,074. You'll want to identify the variables that are needed to calculate the decay and input them in the decay formula.

Continuous Exponential Decay

Continuous decay occurs when the quantity is continuously changing such as the growth of bacteria. Bacteria grow continuously from birth, continuously splitting into halves. Another example is the growth of a tree.

A tree grows at all times, whether we are aware of it or not. Such continuous growth cannot be properly captured by the earlier formula for non-continuous decay. The decay formula for continuous decay is presented like this:

y = Ae-kt    

Where

• y is the remaining quantity after the decay
• A is the initial amount of the quantity before the starting of the decay.
• -k is a constant that represents the rate of decay. The k is negative because it decays. It would be positive for exponential growth.
• t is the time
• e is called Euler’s number and is equivalent to the irrational number 2.718281828459045… which is approximated to 2.718. e is used because it happens to be a number that describes a lot of natural phenomena.

Example 2

If you've never heard of radioactive elements, they're elements or materials that decay with time into other elements. For example, if you leave a sample of Uranium for a while, it will decay to thorium. The decay is a continuous process.

Let’s say you have 20g of a radioactive element, A. After an hour, 7% of the radioactive element A decayed to another element B. How many grams of element A will you have after 8 hours?

Step 1

The first step is to write our continuous decay formula.

y = Ae-kt

Let’s write the data that is available from the question.

• A = 20 which is the initial amount
• e = 2.718. Note that your calculator knows the value of e, so you don’t have to write it in numbers.
• k = 7% = 0.07 which is the rate of decay
• t = 8 which is the number of time intervals.

Step 2

The second step is to substitute the values of the variables in the formula and calculate.

• y = 20 * e-0.07 * 8
• y = 20 * e-0.56
• y = 20 * 0.57
• y = 11.424

So at the end of 8 hours, you will have 11.424g of element A remaining.

Example 3

You work in a radioactive lab where you handle different types of radioactive elements. In the lab, there's another radioactive element, C, which has decayed to half of its mass in 24 hours. You want to know the decay rate of element C. Unfortunately, you can’t find the decay chart that you normally use. So you have to calculate it.

Step 1

Write your decay formula:

y = Ae-kt

As usual, let us write the data that is available from the question. Your initial amount, A is not given. But you are told that at the end of 24 hours, the mass is reduced to half. This means that the mass you are left with is ( ½ )A.

The time given, t, is 24 hours, and the decay rate, k, is what we are looking for.

Step 2

Substitute the values of the variables in the formula and calculate the decay rate.

• ½ A = Ae-k * 24
• ½ A = Ae-24k
• ½ = e-24k

Since A is on both sides of the equation, we can eliminate it by dividing both sides of the equation by A.

• Take the natural log of both sides: ln ( ½ ) = ln ( e-24k )
• Remember that ln ( ex ) = x, therefore ln ( ½ ) = -24k
• Divide both sides by -24: [( ln ½ ) / -24 = k], [-0.693 / -24 = k]
• k = 0.029

So the rate of decay of element C is 0.029 or 2.9%.

Calculating Your Exponential Decay

When it comes to exponentials, you have to first determine if it's a continuous decay or non-continuous, and then apply the appropriate formula. If the quantity changes without any breaks, use the continuous decay formula: y = Ae-kt

However, if the process changes only at the end of a particular period, then use the non-continuous formula: y = a ( 1 – r )x

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