Problem 36 - Applied Calculus

Question

The gross world product is \(W=32.4(1.036)^{t}\), where \(W\) is in trillions of dollars and \(t\) is years since \(2001 .\) Find a formula for gross world product using a continuous growth rate.

Step-by-Step Solution

Answer

The continuous growth formula is \( W = 32.4 e^{0.03537t} \).

Step 1: Understand the Given Formula

The formula provided is an exponential growth equation in the form \( W = 32.4(1.036)^t \), where \( W \) is the gross world product in trillions of dollars and \( t \) is the number of years since 2001. Here, the base of the exponent, 1.036, indicates a discrete growth factor.

Step 2: Convert to Continuous Growth Formula

To express the growth in terms of a continuous growth rate, we need to rewrite the formula using the base of the natural logarithm, \( e \). The general formula for continuous growth is \( W = W_0 e^{rt} \), where \( r \) is the continuous growth rate.

Step 3: Use Base Conversion

We know that \( 1.036 \equiv e^{r} \). Find \( r \) by solving the equation \( e^{r} = 1.036 \) using natural logarithms: \( r = \ln(1.036) \).

Step 4: Calculate the Continuous Growth Rate

Compute \( r \) as follows: \( r = \ln(1.036) \approx 0.03537 \). This means the continuous growth rate is approximately 3.537% per year.

Step 5: Write the Continuous Growth Formula

Substitute \( r \) into the continuous growth formula. The formula becomes \( W = 32.4 e^{0.03537t} \).

Step 6: Interpret the Formula

The new formula \( W = 32.4 e^{0.03537t} \) approximates the gross world product, showing that it grows continuously at an annual rate of approximately 3.537% since 2001.

Key Concepts

Continuous Growth Rate

When we talk about continuous growth rate, we are referring to a type of growth that happens smoothly and steadily over time. Unlike discrete growth, which might happen in sudden jumps or intervals, continuous growth is ongoing. The exponential function describes this type of growth perfectly and uses natural exponential base, represented by the symbol \( e \). Here’s why continuous growth is useful in real-world scenarios:

• It accounts for growth that happens consistently over each moment, not just at fixed intervals.
• It is useful in financial models, biology, and any field where a steady, uninterrupted growth pattern is expected.

To convert a discrete growth rate to a continuous growth rate, we use natural logarithms. Solving for our continuous rate involves finding the equivalent base \( e \) form. If given a percentage growth in a discrete scenario, you can utilize \( e^{r} \), where \( r \) will represent the continuous rate of growth. This conversion is essential for understanding how variables like gross world product evolve smoothly over time.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. This logarithm plays a crucial role in calculations involving continuous growth because it helps us make sense of exponential changes by linearizing them. Here’s a breakdown of why it’s helpful:

• It allows us to determine the continuous growth rate from a known discrete growth factor.
• It simplifies the exponential expressions, making them easier to interpret and manipulate.
• Natural logarithms provide a direct way to solve for time variables in exponential equations.

In our exercise, converting from a base of 1.036 to \( e^{r} \), involved using the natural logarithm, \( r = \ln(1.036) \). This step is essential to shift from a non-continuous growth rate to a continuous framework. The magic of natural logarithms lies in their ability to transform multiplication into addition, simplifying complex exponential growth narratives.

Exponential Functions

Exponential functions are mathematical expressions that model growth or decay processes. They are characterized by the exponent in the formula, which signifies a rapid change. In the context of our world product growth, an exponential function can be expressed as \( W = W_0 e^{rt} \). Here's why exponential functions are significant:

• They naturally model phenomena where change accelerates over time, like population growth or financial investments.
• The base of the exponential function tells us about the growth factor. In discrete models, it might be a number like 1.036, but for continuous processes, it is the constant \( e \).
• They allow for predictions into future values by understanding both the initial quantity \( W_0 \) and the growth parameter \( r \).

In the continuous growth formula \( W = 32.4 e^{0.03537t} \), the exponent \( rt \) drives how rapidly the world's gross product grows. Exponential functions bring together constants like \( e \), initial values, and growth rates to produce a model that predicts ongoing change over time efficiently.