Problem 31 - Applied Calculus

Question

(a) What is the continuous percent growth rate for \(P=\) \(100 e^{0.06 t},\) with time, \(t,\) in years? (b) Write this function in the form \(P=P_{0} a^{t} .\) What is the annual percent growth rate?

Step-by-Step Solution

Answer

(a) Continuous growth rate: 6%. (b) Annual growth rate: ~6.18%.

Step 1: Identify Continuous Growth Rate

The given function is \(P = 100 e^{0.06t}\). This is in the form of \(P = P_0 e^{rt}\), where \(r\) is the continuous growth rate. Here, \(r=0.06\). Therefore, the continuous percent growth rate is \(6\%\).

Step 2: Convert to Exponential Form

We need to express \(P = 100 e^{0.06t}\) in the form \(P = P_0 a^{t}\). The equivalent of the exponential base \(e^{0.06}\) is needed. We know \(e^{0.06}\) equals \(a\). Calculate \(a\) to be \(e^{0.06} \approx 1.0618\).

Step 3: Calculate Annual Percent Growth Rate

After converting to the form \(P = P_0 a^{t}\), \(a = 1.0618\) represents the annual growth multiplier, where \(a = 1 + \text{growth rate}\). Hence, \(\text{growth rate} = a-1 = 1.0618 - 1 = 0.0618\), which translates to an annual percent growth rate of approximately \(6.18\%\).

Key Concepts

Continuous Growth Rate

In mathematical modeling, the continuous growth rate is fundamental for understanding exponential growth behaviors. It's often used in contexts where quantities increase smoothly over time. The given exponential model, \( P = 100 e^{0.06t} \), illustrates continuous growth. Here, the variable \( t \) represents time in years, and \( P \) is the quantity at a given time. The term \( e^{0.06t} \) shows the nature of continuous growth.

The continuous growth rate is the proportion by which a quantity grows continuously. For the expression \( P = P_0 e^{rt} \), \( r \) directly represents this continuous rate, expressed as a decimal. In our example, \( r = 0.06 \), indicating a continuous growth rate of \( 6\% \).

When you see an expression with the base \( e \), it signals that the growth is not discrete or spaced in time but is occurring continuously. This specific base, known as Euler's number, is approximately \( 2.71828 \). As such, exponential growth linked to \( e \) is constantly compounding, making it essential for precise models in population studies and finance.

Annual Percent Growth Rate

The annual percent growth rate allows us to perceive how much a quantity increases in one complete year. While continuous growth rates are useful in theoretical analysis, annual rates often provide a more intuitive grasp of growth for practical scenarios.

To find the annual percent growth rate, we must first transition the equation from its continuous form \( P = 100 e^{0.06t} \) into the discrete exponential form \( P = P_0 a^t \). As this involves calculating the equivalent of \( e^{0.06} \), we find \( a = e^{0.06} \approx 1.0618 \).

This number, \( a \), is the annual growth factor, where \( a = 1 + \text{growth rate} \). Therefore, \( 1.0618 \) signifies an annual growth rate of \( 0.0618 \) or \( 6.18\% \). This percentage provides a clearer view of yearly changes and is typically used when formulating expectations over set periods.

Exponential Functions

Exponential functions like \( P = P_0 e^{rt} \) or \( P = P_0 a^t \) are versatile in depicting rapid growth patterns across numerous fields. These functions show how quantities grow proportional to their current size, exemplifying natural growth behaviors in populations and investments.

An exponential function consists of several elements:

• Initial Amount \( P_0 \): Tradition sets this as the starting value against which all growth is measured.
• Growth Factor: In continuous models, represented by \( e^{rt} \); in discrete, by \( a^t \).
• Rate \( r \) or growth increase multiplier \( a \): These indicate how quickly growth happens.

The superb aspect of exponential functions is their capacity for illustrating compounded growth, where increases build over previous gains, leading to swift escalations. Predictably, they form the backbone of models analyzing trends in economics, demographics, and natural sciences, due to their capability to capture progressive changes.