Exponential Growth Formula Derivation and Solved Examples
Before answering what is exponential growth? Let us know, why is it called exponent? An exponent refers to the number of times a number is multiplied to itself. We can have a clear idea on the topic by solving exponential growth equations.
Something which always grows in relation to the current value is known as exponential growth. Exponential growth is also known as doubling the existing number. Let us take an example: If the population of rabbits grows every month, then we would have 2, then 4, 8, 16, 32, 64, 128, 256, and further carried on. The formula used in solving exponential growth equations is y = abx.
Exponential Growth Graph
The general shape of an exponential growth graph is ‘J’ shape. This shows that the graph is always increasing in nature. An exponential growth graph is drawn using the function of the form y = abx where, a > 0 and b > 1. Note that the graph shoots upward rapidly as ‘X’ increases. This is because of the doubling behavior of the exponential. Below given graph is an example of exponential growth.
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Exponential Growth Equation
For solving exponential growth equations you need to have equations with an exponential expression on both sides of the equal sign. Then compare the powers and solve the given equation. Let us solve one basic problem to understand it better.
Q. Solve 51-x = 54
As the bases are the same we can equate the powers and solve the problem.
1 - x = 4
x = -3
This is the required solution to the above problem.
It is used in solving exponential growth equations when there is a comparison between two cases.
Exponential Decay
Something which ‘Decay’(gets smaller) with respect to time exponentially is known as exponential decay. Exponential decay is applied when anything decreases compared to the current value exponentially. For example, we can take atmospheric pressure of the air around us which decreases as we go up. It decreases by about 12% for every 1000m which is exponential decay.
Exponential Decay Formula
The formula used to express the exponential decay is y = a(1 - b)x.
Where y is the final amount, a is the original amount, b is the decay factor, and x the amount of time that has passed. Exponential decay formula is useful in solving a variety of real-world problems, Most popular application is to track inventory that is used regularly in the same quantities.
Solved Problems
Q. The population of a city in 2016 estimated to be 35,000 people with an annual rate of increase of 2.4%.
Solution: Now let us find the growth factor,
After one year population would be = 35,000 + 0.024(35,000)
By factoring above equation becomes = 35,000 (1 + 0.024)
The growth factor is b = 1.024 (Remember that it is greater than 1)
Now, The general formula for exponential growth is y = abx
Substituting the value in above formula y = 35,000(1.024)x
Consider, that we are using this estimate of the population in 2020 to the nearest hundred people.
Y = 35,000 (1.024)4≈ 38,482.91 ≈ 38,500
So, the estimated population of the city in 2020 is 38,500.
FAQs on Exponential Growth Concept and Mathematical Model
1. What is exponential growth in maths?
Exponential growth is a type of increase where a quantity grows by a constant percentage rate over equal time intervals. In exponential growth, the amount multiplies rather than increases by a fixed addition.
- General form: y = a(1 + r)t
- a = initial value
- r = growth rate (as a decimal)
- t = time periods
2. What is the formula for exponential growth?
The formula for exponential growth is y = a(1 + r)t. This formula models situations where growth happens at a fixed percentage rate.
- a = initial amount
- r = growth rate per period
- t = number of time periods
- y = final amount
3. How do you calculate exponential growth step by step?
To calculate exponential growth, substitute the known values into the formula y = a(1 + r)t and simplify.
- Step 1: Identify a, r, and t.
- Step 2: Convert percentage to decimal (e.g., 5% = 0.05).
- Step 3: Substitute into the formula.
- Step 4: Evaluate the exponent.
4. What is an example of exponential growth?
An example of exponential growth is compound interest where money grows by a fixed percentage each year. Suppose $500 is invested at 8% annually for 2 years.
- Formula: y = 500(1.08)2
- Calculation: 500 × 1.1664 = 583.2
5. What is the difference between exponential growth and linear growth?
The key difference is that exponential growth increases by a constant percentage, while linear growth increases by a constant amount.
- Linear growth: y = a + bt (adds the same value each time)
- Exponential growth: y = a(1 + r)t (multiplies by the same factor each time)
6. How do you know if a function shows exponential growth?
A function shows exponential growth if the variable appears in the exponent and the base is greater than 1. The general form is y = abt where b > 1.
- The rate of increase gets faster over time.
- The graph curves upward.
- Successive values have a constant ratio.
7. What is the doubling time in exponential growth?
Doubling time is the time it takes for a quantity undergoing exponential growth to become twice its initial value. It can be calculated using T = ln(2) / ln(1 + r).
- ln means natural logarithm.
- r is the growth rate (decimal).
8. Where is exponential growth used in real life?
Exponential growth is used to model situations where quantities grow rapidly at a constant percentage rate. Common real-life applications include:
- Compound interest in finance
- Population growth
- Bacterial growth
- Spread of viruses
9. What happens to the graph of an exponential growth function?
The graph of an exponential growth function curves upward and becomes steeper over time. For y = a(1 + r)t where r > 0:
- The graph passes through (0, a).
- It increases slowly at first.
- It rises rapidly as t increases.
10. What is the growth factor in exponential growth?
The growth factor is the number multiplied each time period and is equal to (1 + r). It determines how quickly the quantity increases.
- If r = 0.05, growth factor = 1.05.
- If r = 0.2, growth factor = 1.2.
