Problem 34 - 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

Continuous growth rate: 6%; Annual growth rate: 6.18%.

Step 1: Identify the Function

The given function is \( P = 100 e^{0.06 t} \), where the base of the exponential function is \( e \) and the exponent is \( 0.06t \).

Step 2: Continuous Percent Growth Rate

The equation \( P = 100 e^{0.06 t} \) shows continuous growth at a rate determined by the coefficient of \( t \) in the exponent. This is directly the continuous growth rate. Thus, the continuous percent growth rate is 6%. This is found because \( 0.06 \) as a decimal converts to 6%.

Step 3: Convert to Annual Growth Form

Convert the function into the form \( P = P_0 a^t \). We know \( a = e^{0.06} \). This involves using properties of exponents.

Step 4: Calculate \( a \)

Use a calculator to find \( e^{0.06} \). The value is approximately \( a \approx 1.0618 \).

Step 5: Calculate Annual Growth Rate

The annual growth rate is found by subtracting 1 from \( a \) and converting to percentage: \( (1.0618 - 1) \times 100 \approx 6.18\% \).

Step 6: Conclusion

The continuous percent growth rate is 6%, and the annual percent growth rate is approximately 6.18%.

Key Concepts

Continuous Growth Rate

In exponential functions involving continuous growth, the term "continuous growth rate" is vital. Here, the function is expressed in the form of \[ P = P_0 e^{rt}, \] where:

• \( P_0 \) is the initial amount.
• \( e \) is the base of the natural logarithm, approximately 2.718.
• \( r \) is the continuous growth rate as a decimal.
• \( t \) is time.

In our example, the function is given as \( P = 100e^{0.06t} \). Here, \( 0.06 \) is the coefficient of \( t \) in the exponent, representing the continuous growth rate. To find this in percent, convert \( 0.06 \) to a percentage by multiplying by 100, resulting in a continuous growth rate of 6%. This means that the quantity is continuously growing at 6% per unit of time.

Annual Growth Rate

The annual growth rate is another way to represent growth over time, but specifically assessed on a yearly basis. Unlike continuous growth, it breaks down the exponential increase into a format that is more relatable as it parallels one full year. To convert a function from continuous growth to annual growth rate format, the equation is transformed into \[ P = P_0 a^t, \] where:

• \( a \) is the annual growth factor.

In our exercise, the task was to express \( P = 100 e^{0.06 t} \) as \( P = 100a^t \). The value of \( a \) is determined by calculating \( e^{0.06} \). Upon calculation, \( e^{0.06} \) results in approximately 1.0618. Therefore, \( a \approx 1.0618 \), making the annual growth rate \((1.0618 - 1)\times100\approx 6.18\%\). While slightly higher than the continuous rate, this value helps understand the overall growth experienced in one year in more practical terms.

Exponential Functions

Exponential functions are a fundamental aspect of understanding growth processes where the rate of change is proportional to the current value. They take the form \[ P = P_0 \, b^t, \] or when dealing with continuous growth, as \[ P = P_0 e^{rt}. \]Their key features include:

• A constant proportional growth rate.
• Exponential increase or decrease depending on the rate sign.
• Non-linear growth which can lead to very rapid changes over periods of time.

The exercise used the exponential function \( P = 100 e^{0.06 t} \) to depict continuous growth. When expressed with the base \( e \), they conveniently show natural growth processes like population and interest accrual. While exponential functions can initially be complex, dissecting them into individual components like base, exponent, and growth types makes them easier to grasp. Understanding both continuous and annual growth within exponential functions provides a comprehensive view of how different representation methods offer insights into the time-related behavior of quantities.