Problem 38 - 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 formula is \( W = 32.4e^{0.0353t} \).

Step 1: Understand the Given Formula

The gross world product is given as \( W = 32.4(1.036)^t \). This is an exponential growth formula where 32.4 is the initial value when \( t = 0 \), corresponding to the year 2001, and 1.036 is the growth factor for each year.

Step 2: Identify the Relationship to Continuous Growth

To convert the given discrete growth rate to a continuous growth rate, we use the relationship \( a = e^r \), where \( a \) is the growth factor (1.036) and \( r \) is the continuous growth rate we need to find.

Step 3: Solve for Continuous Growth Rate \( r \)

Using the equation \( 1.036 = e^r \), we take the natural logarithm on both sides to solve for \( r \). Thus we have \( r = \ln(1.036) \).

Step 4: Compute \( r \) Using Natural Logarithm

Calculate \( r \) by evaluating \( \ln(1.036) \). Using a calculator, \( \ln(1.036) \approx 0.0353 \).

Step 5: Write the Formula with Continuous Growth Rate

Using the continuous growth rate \( r \), the formula for gross world product becomes \( W = 32.4e^{0.0353t} \). This formula represents an equivalent model with continuous growth.

Key Concepts

Continuous Growth Rate

Continuous growth rate is a concept used when growth happens in an unbroken manner over time. In the context of exponential growth, it signifies the rate at which a quantity grows continuously.

The continuous growth rate, often represented by the symbol 'r', allows us to describe situations where growth occurs in an uninterrupted and seamless fashion. Unlike discrete growth, where changes happen at specific intervals, continuous growth assumes that small bits of growth are added at every instant of time.

• To transition from a discrete growth model to a continuous one, you typically use the formula relating the discrete factor to the continuous rate:
  - Discrete growth factor (a) = 1.036- Continuous rate (r) can be found with the equation: \[ a = e^r \]

Understanding continuous growth rates is essential because many natural processes, such as populations or investments, often grow continuously rather than in spurts. In this specific exercise, the continuous growth rate is determined by converting the discrete growth factor using the natural logarithm.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm that uses the natural base \( e \), approximately equal to 2.71828. It is ubiquitous in calculations involving continuous growth rates.

The natural logarithm helps to easily transition between exponential forms, especially in solving equations like \( a = e^r \).

• In solving exponential growth problems, it is often necessary to take the natural logarithm of both sides of the equation to isolate the continuous growth rate \( r \).

Let's look at an example from the exercise:

• Given: \(1.036 = e^r\)
• Taking the natural logarithm leads to: \(\ln(1.036) = r\)

Calculators or computer software can easily compute natural logarithms, providing us with results like \( \ln(1.036) \approx 0.0353 \), which help in forming the continuous growth model.

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They have the general form \( y = ab^t \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( t \) represents time.

Exponential functions are prominent in modeling growth processes because they can precisely represent behavior that grows by a constant proportion over equal time intervals.

• In the world of continuous growth, the formula often reformulates to \( y = ae^{rt} \). - Here, \( e \) is the base of the natural logarithm, and \( r \) is the continuous growth rate.

For the specific problem of gross world product:

• Original formula with discrete growth: \( W = 32.4(1.036)^t \)
• Equivalent formula with continuous growth: \( W = 32.4e^{0.0353t} \)

This transformation shows how exponential functions can adapt to different growth scenarios, providing a versatile tool for analyzing change over time.