Problem 20 - Applied Calculus

Question

Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=3.2 e^{0.03 t}$$

Step-by-Step Solution

Answer

Initial quantity is 3.2, growth rate is 3%, and it is continuous.

Step 1: Identify the General Formula

The function given is in the form \( P = P_0 e^{kt} \). This is the standard formula for exponential growth or decay, where \( P_0 \) is the initial quantity, \( k \) is the growth rate constant, and \( t \) is time.

Step 2: Identify the Initial Quantity

From the formula \( P = P_0 e^{kt} \), we see that \( P_0 \) is the initial quantity. In the given equation, this is \( 3.2 \). Thus, the initial quantity is 3.2.

Step 3: Determine the Growth Rate

In the given expression \( P = 3.2 e^{0.03 t} \), the constant \( 0.03 \) in the exponent of \( e \) represents the growth rate constant \( k \). This means the continuous growth rate is 3%.

Step 4: Check the Nature of the Growth Rate

Since the equation is of the form \( P = P_0 e^{kt} \) and uses the exponential function \( e \), the growth described is continuous. Thus, the rate is a continuous growth rate.

Key Concepts

Initial Quantity

When dealing with exponential growth, understanding the initial quantity is crucial. The initial quantity is essentially the starting point of whatever is growing or decaying. In formulas for exponential growth or decay, such as \( P = P_0 e^{kt} \), this initial quantity is represented by \( P_0 \). It sets the stage for how the quantity will evolve over time. In the given exercise, you've encountered the equation \( P = 3.2 e^{0.03 t} \). Here, the initial quantity \( P_0 \) is clearly stated as 3.2. This means that whatever you are observing started from 3.2 units. To summarize:

• Initial quantity \( P_0 \) is the value at \( t = 0 \).
• It's vital for establishing the baseline for growth.

Growth Rate

The growth rate in an exponential function indicates how quickly the quantity is increasing or, in some cases, decreasing over time. In the standard exponential growth formula \( P = P_0 e^{kt} \), \( k \) represents the growth rate. It tells us the percentage change per unit time.In our exercise equation, \( P = 3.2 e^{0.03 t} \), the growth rate is represented by the constant \( 0.03 \). This means that the quantity grows by 3% continuously over each time period. Key takeaways:

• \( k = 0.03 \) indicates a 3% continuous growth rate.
• Positive \( k \) values signify growth, while negative values indicate decay.

Exponential Function

The exponential function is a mathematical expression of the form \( f(t) = P_0 e^{kt} \), which efficiently models growth and decay processes. The core of this function is the base of the natural logarithm \( e \), which ensures that the growth or decay is continuous rather than occurring in discrete jumps.The exponential function is versatile and can describe various phenomena, such as population growth, compound interest, or spread of diseases.In this exercise's function \( P = 3.2 e^{0.03 t} \):

• \( e^{0.03 t} \) represents continuous change.
• The form \( e^{kt} \) is indicative of continuous processes by nature.

Understanding how the exponential function works helps in predicting and analyzing how a model evolves over time.