Problem 21 - Applied Calculus

Question

Write the functions in Problems \(21-24\) in the form \(P=P_{0} a^{t}\) Which represent exponential growth and which represent exponential decay? $$P=15 e^{0.25 t}$$

Step-by-Step Solution

Answer

The function represents exponential growth, written as \( P = 15 (e^{0.25})^t \).

Step 1: Identify the Original Function Form

The original function given is \( P = 15 e^{0.25t} \), which follows the form \( P = P_0 e^{kt} \).

Step 2: Recognize Exponential Growth Form

In exponential form \( P = P_0 a^t \), \( a \) must be greater than 1 for exponential growth. Here, \( e^{0.25} \) is constant and positive, indicating growth because \( a = e^{0.25} \) is greater than 1.

Step 3: Rewrite Using Exponential Growth Formula

Rewrite the expression in the exponential form \( P = P_0 a^t \) by equating \( a = e^{0.25} \). Thus, the function becomes \( P = 15 (e^{0.25})^t \).

Step 4: Determine Exponential Behavior

Since \( e^{0.25} > 1 \), the function represents exponential growth.

Key Concepts

Exponential Growth

Exponential growth describes a process where a quantity increases over time by a constant multiplicative rate. This means that as time progresses, the value becomes larger and grows at an ever-accelerating pace.
Let's consider the mathematical definition of exponential growth. The general form is \( P = P_0 a^t \), where:

• \( P \) is the current amount.
• \( P_0 \) is the initial amount.
• \( a \) is the base of the exponential, and it must be greater than 1 for growth.
• \( t \) is the time period considered.

In any exponential growth scenario, \( a \) speaks to how much the quantity grows with each time step. For example, if \( a = 2 \), then the quantity doubles each time period.
Using the exercise example \( P = 15 e^{0.25t} \), we see that it fits this mode of growth because \( e^{0.25} \) exceeds 1. As a result, the factor \( e^{0.25} \) encourages the initial amount 15 to grow to larger and larger values as time proceeds.

Exponential Decay

Exponential decay represents the process where a quantity decreases over time at a constant proportionate rate. This is essentially the opposite of exponential growth, resulting in the quantity shrinking down rapidly.
In mathematics, exponential decay can be expressed using the equation form \( P = P_0 a^t \), quite similar to that of exponential growth but with a slight twist:

• \( a \) in decay must be between 0 and 1, indicating decline with time.

This decrease implies that with each time step, the quantity reduces by a consistent percentage, thus losing its total volume exponentially.
Take, for example, a radioactive substance decaying; if \( a \) is 0.5, the substance halves in amount with each time period. While the exercise does not directly involve decay, understanding decay helps in contrasting and grasping the differentiation between growth and decay scenarios.

Mathematical Functions

Mathematical functions represent relationships between sets of numbers or values, providing frameworks through which we can understand and predict patterns within different contexts.
A function is usually defined by its input-output behavior, typically noted as \( f(x) \) for a given input \( x \). They link each input to exactly one output, acting like machines that process data predictably.
In exponential functions like \( P = P_0 a^t \), they serve to outline how the output (\( P \)) changes with time (\( t \)), based on the initial state (\( P_0 \)) and the growth or decay factor (\( a \)). These functions are foundational in various fields such as finance for modeling compound interest, biology for population dynamics, and physics for decay processes.
Understanding mathematical functions, particularly exponential ones, equips you with the tools to analyze changes involving growth or decay, providing insights and predictions about dynamic systems across multiple areas.