Exponential Growth
Exponential growth is a specific way that a quantity may increase over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself.
Mathematical Model
A typical model for exponential growth is given by the equation:
A = P * e(rt)
Where:
- A represents the amount after time t
- P is the initial amount (Principal)
- r is the rate of growth
- t is the time
- e is Euler's number, approximately equal to 2.71828
Applications in Population Growth
In biology, exponential growth is used to model population growth under the assumption of unlimited resources and absence of other limiting factors like predation, diseases, and competition for resources.
Example 1: Bacterial Growth
Consider a bacterial population that doubles every hour. If you start with one bacterium, how many bacteria will there be after 6 hours?
Solution: Using the formula: A = 1 * 26 = 64 There will be 64 bacteria after 6 hours.
Applications in Financial Growth
In finance, exponential growth models are used to describe the growth of investments under compound interest.
Example 2: Investment Growth
Suppose you invest £1000 at an annual interest rate of 5%. How much will you have after 10 years?
Solution: Using the formula: A = 1000 * e(0.05 * 10) Calculating it out: A = 1000 * e0.5 A ≈ 1000 * 1.64872 A ≈ £1648.72
Exponential Decay
Exponential decay describes a decrease at a rate that’s proportional to the current value. It is often used to model natural phenomena like radioactive decay or cooling/heating processes.
Mathematical Model
The mathematical model for exponential decay is:
A = P * e(-rt)
Applications in Radioactive Decay
Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. However, the chance that a given atom will decay is constant over time.
Example 3: Radioactive Element Decay
If you have 100g of a radioactive element with a half-life of 3 years, how much will remain after 9 years?
Solution: Using the formula: A = 100 * e(-0.231 * 9) Calculating it out: A = 100 * e-2.079 A ≈ 100 * 0.125 A ≈ 12.5 So, 12.5g of the element will remain after 9 years.
Applications in Newton’s Law of Cooling
Newton’s law of cooling states that the rate of change of the temperature T of an object is proportional to the difference between its own temperature and the ambient temperature (the temperature of its surroundings).
Example 4: Cooling of a Hot Liquid
Suppose a cup of coffee is left to cool in a room with a constant temperature of 20°C and it cools from 95°C to 75°C in 2 minutes. How long will it take to cool from 75°C to 55°C?
Solution: To solve the problem, we can use Newton's Law of Cooling, which states that the rate of change of the temperature (T) of an object is proportional to the difference between the temperature (T) of the object and the ambient temperature (Troom). Mathematically, this can be expressed as:
dT/dt = -k(T - Troom)
Given that the coffee cools from 95°C to 75°C in 2 minutes, we can use this information to find the value of k. Then, we can use k to find how long it will take for the coffee to cool from 75°C to 55°C.
Let's find the value of k first. We know that: T_room = 20°C T1 = 95°C T2 = 75°C Delta t1 = 2 minutes
We can rearrange Newton's Law of Cooling to find k: k = -1/Delta t1 * ln((T2 - Troom)/(T1 - Troom))
Calculating the value of k, we get approximately 0.155.
Now, we can use this value of k to find how long it will take for the coffee to cool from 75°C to 55°C. We'll use the same formula, but rearrange it to solve for time t:
t = (T/k) * ln((T1 - Troom)/(T2 - Troom))
Where: T1 is the initial temperature (75°C). T2 is the final temperature (55°C). Troom is the ambient temperature (20°C). k is our constant (0.155).
Calculating the time t, it will take for the coffee to cool from 75°C to 55°C, we get approximately 8.75 minutes.
FAQ
Exponential decay models describe phenomena that decrease at a rate proportional to their current value, such as radioactive decay or depreciation of assets. Mathematically, decay models typically involve bases between 0 and 1 (0 < b < 1). Conversely, exponential growth models describe phenomena that increase at a rate proportional to their current value, like population growth or investment growth. In these models, the base is usually greater than 1 (b > 1). The key difference lies in the direction of change—decay models decrease towards zero, while growth models increase without bound.
Exponential functions are widely used in financial models to represent compound interest, which is a fundamental concept in finance. The formula A = P * e(rt) is used to calculate the future value (A) of an investment (P) growing continuously at a rate (r) for a time (t). This model is applied in various financial scenarios, such as calculating the future value of investments, loans, and mortgages, providing a realistic and applicable model for continuous growth or decay in a financial context.
While exponential functions are widely applicable and provide accurate models in numerous real-world scenarios, they are not universal. Some phenomena initially follow exponential growth or decay but level off over time, which is better modeled with a logistic function. In other instances, growth or decay may follow a polynomial, linear, or another non-exponential model. Therefore, while exponential functions are powerful and commonly used, it's crucial to select a model that accurately reflects the specific characteristics of the phenomenon being studied.
The number 'e' (approximately 2.71828) is a mathematical constant that is the base of the natural logarithm. It's unique in that the function ex has the same slope as its value at every point, making calculations particularly smooth and predictable in calculus. In the context of exponential models, 'e' is often used in continuously compounding interest, population growth, and decay models due to its desirable mathematical properties, such as yielding simple derivatives and having a natural growth rate of 100%.
The base of an exponential function significantly influences its graph. If the base is greater than 1, the function shows exponential growth, meaning it increases as x increases, and it decreases as x decreases, always staying above the x-axis (y > 0). If the base is between 0 and 1 (0 < b < 1), the function exhibits exponential decay, meaning it decreases as x increases and increases as x decreases, but still always remains above the x-axis. The base also affects the steepness of the graph; larger bases result in steeper graphs in the case of exponential growth.
Practice Questions
To solve this question, we can use the exponential growth formula:A = P * 2(t/T)
Where:
- A is the amount after time t
- P is the initial amount
- t is the elapsed time
- T is the doubling period
Substituting the given values:
A = 500 * 2(12/3)
A = 500 * 24
A = 500 * 16
A = 8000
So, after 12 hours, there will be 8000 bacteria. Understanding the relationship between the doubling period and the elapsed time in the context of exponential growth allows us to predict future population sizes based on the current population and growth rate.
To find the half-life, we can use the exponential decay formula:A = P * e(-rt)
Where:
- A is the amount after time t
- P is the initial amount
- r is the decay rate
- t is the time
- e is Euler's number, approximately equal to 2.71828
First, we need to find the decay rate (r) using:r = - (ln(A/P) / t)Substituting the given values:r = - (ln(50/200) / 8)r = - (ln(0.25) / 8)r = - (-1.386294361 / 8)r = 0.1732867951
Next, we find the half-life (T) using:T = ln(2) / rT = 0.693147181 / 0.1732867951T ≈ 4
So, the half-life of the element is approximately 4 years. Understanding the decay rate and its relationship with half-life is essential in predicting the remaining quantity of a substance after a given period, which has applications in fields like archaeology, nuclear physics, and medicine.
