Definition of the exponential growth and decay model (dy/dt = ky)

📚 Definition of the Exponential Growth and Decay Model

The exponential growth and decay model describes the rate of change of a quantity as being proportional to its current value. Mathematically, it is represented by the differential equation:

$\frac{dy}{dt} = ky$

Where:

• 📈 y is the quantity at time t.
• ⏰ t is time.
• 🔑 k is the constant of proportionality; positive for growth, negative for decay.

📜 History and Background

The concept of exponential growth and decay has been around for centuries, finding its roots in early studies of population growth and compound interest. The formal mathematical model was developed as calculus became more sophisticated, enabling precise descriptions of continuous change.

• 🌱 Early studies of population growth by Thomas Malthus in the late 18th century highlighted exponential increases.
• 🏦 Compound interest calculations provided practical examples of exponential growth in finance.
• 🔬 Scientific advancements in the 19th and 20th centuries applied the model to areas like radioactive decay and chemical kinetics.

💡 Key Principles

Understanding the exponential growth and decay model involves grasping a few key principles:

• ➕ Initial Value: The initial amount of the quantity, denoted as $y_0$ or $y(0)$, significantly influences the future values.
• 🧮 Constant of Proportionality (k): The value of k determines whether the quantity grows (k > 0) or decays (k < 0), and the speed of this change.
• ⏳ Time Dependence: The quantity changes continuously over time, following an exponential curve.
• 📐 Solution to the Differential Equation: The general solution to the differential equation $\frac{dy}{dt} = ky$ is given by: $y(t) = y_0e^{kt}$, where e is the base of the natural logarithm.

🌍 Real-world Examples

The exponential growth and decay model appears in numerous real-world scenarios:

• 🦠 Bacterial Growth: Under ideal conditions, a population of bacteria can double at regular intervals, exhibiting exponential growth.
• ☢️ Radioactive Decay: Radioactive isotopes decay at a rate proportional to their amount, characterized by a half-life.
• 💰 Compound Interest: The value of an investment grows exponentially when interest is compounded continuously.
• 🌡️ Cooling/Heating: Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its own temperature and the ambient temperature.

🧪 Example Problems

Let's explore some examples.

1. Bacterial Growth:

   A bacteria culture starts with 500 cells. It grows at a rate proportional to its size. After 3 hours, there are 8000 cells. Find the expression for the number of cells after $t$ hours.

   Solution:

   $\frac{dy}{dt} = ky$, $y(0) = 500$, $y(3) = 8000$.

   $y(t) = 500e^{kt}$

   $8000 = 500e^{3k}$

   $16 = e^{3k}$

   $k = \frac{ln(16)}{3}$

   $y(t) = 500e^{(\frac{ln(16)}{3})t}$

2. Radioactive Decay:

   The half-life of a radioactive substance is 1500 years. If a sample has a mass of 100mg, find the mass remaining after $t$ years.

   Solution:

   $\frac{dy}{dt} = ky$, $y(0) = 100$, $y(1500) = 50$.

   $y(t) = 100e^{kt}$

   $50 = 100e^{1500k}$

   $\frac{1}{2} = e^{1500k}$

   $k = \frac{ln(0.5)}{1500}$

   $y(t) = 100e^{(\frac{ln(0.5)}{1500})t}$

🎯 Conclusion

The exponential growth and decay model is a powerful tool for understanding phenomena in various fields, from biology to finance. Its simplicity and broad applicability make it an essential concept in mathematical modeling.