Problem 46 - Calculus

Question

The third-quarter revenue of Apple(R) went from \(\$ 3.68\) billion \(^{30}\) in 2005 to \(\$ 15.68\) billion \(^{31}\) in \(2010 .\) Find an exponential function to model the revenue as a function of years since \(2005 .\) What is the continuous percent growth rate, per year, of sales?

Step-by-Step Solution

Answer

The continuous percent growth rate of Apple's revenue is approximately 33.62% per year.

Step 1: Identify the Exponential Growth Model

We know that the revenue follows the exponential growth model: \( R(t) = R_0 e^{kt} \), where \( R(t) \) is the revenue at time \( t \), \( R_0 \) is the initial revenue, and \( k \) is the continuous growth rate. Our task is to find \( k \).

Step 2: Set Initial Conditions

From the problem, we know the initial revenue in 2005, which is \( R_0 = 3.68 \) billion dollars. This is our revenue when \( t = 0 \).

Step 3: Use Given Data for 2010

In 2010, which is 5 years after 2005, the revenue is \( R(5) = 15.68 \) billion dollars. We can substitute these values into the exponential model: \( 15.68 = 3.68 e^{5k} \).

Step 4: Solve for the Growth Rate

To find \( k \), divide both sides by \( 3.68 \): \( \frac{15.68}{3.68} = e^{5k} \). Calculate the left side: \( 4.26 = e^{5k} \).

Step 5: Isolate the Exponential Term

Take the natural logarithm of both sides to solve for \( k \): \( \ln(4.26) = 5k \).

Step 6: Calculate the Continuous Growth Rate

Isolate \( k \) by dividing both sides by 5: \( k = \frac{\ln(4.26)}{5} \). Compute \( k \) using a calculator.

Step 7: Interpret the Result

The value of \( k \) represents the continuous growth rate per year of Apple's third-quarter revenue from 2005 to 2010.

Key Concepts

Continuous Growth Rate

When analyzing the growth of anything over time, it can be helpful to determine the continuous growth rate. It tells us how quickly something is growing continuously and can apply to various contexts, such as populations, investments, or company revenues.

In the case of Apple's revenue from 2005 to 2010, we're looking for the continuous growth rate, denoted as \( k \) in our exponential model equation \( R(t) = R_0 e^{kt} \). Here, \( R(t) \) is the revenue at time \( t \), and \( R_0 \) is the revenue at the starting point, which is 2005 in this scenario. The continuous growth rate \( k \) helps explain how much the revenue has compounded within that timeframe.

To find the continuous growth rate, you'd follow these steps:

• Understand the exponential growth model and identify \( R_0 \), the initial revenue figure.
• Insert the known values for revenue over the timeframe into the equation to solve for \( k \).
• Take the natural log of the resulting quotient to find \( k \).

The resulting \( k \) value gives us a powerful insight: it shows how the revenue would increase if it grew at a constant percentage rate each year without any interruptions.

Exponential Function

An exponential function is foundational for modeling situations where growth accelerates over time, such as compound interest or population growth. The core idea of an exponential function is that some quantity increases by a fixed percentage over equal time periods.

For the Apple revenue scenario, we can express the exponential function as \( R(t) = R_0 e^{kt} \). This formula helps predict how revenue grows exponentially rather than linearly over time.

• \( R(t) \) represents the revenue at some time \( t \) years since the starting year (2005 in this case).
• \( R_0 \) is the initial revenue amount —\( 3.68 \) billion dollars here.
• \( k \) stands for the continuous growth rate.
• \( e \) is the base of natural logarithms, roughly equal to 2.71828.

The strength of the exponential function lies in its ability to genuinely represent real-world situations where increases become greater over time, often initially appearing quick then exploding in size due to compounding effects.

Revenue Modeling

Modeling revenue with mathematical functions is crucial when making predictions or analyzing business growth. An exponential growth model represents scenarios where revenue doesn’t just increase periodically, but compounds continuously.

For an existing business, accurately modeling revenue is vital for strategic planning and investment decisions. By using the exponential revenue model, a business like Apple forecasts how past trends might influence future revenue streams.

Revenue modeling helps businesses:

• Understand how past growth might predict future opportunities.
• Determine what investments might be necessary to sustain growth.
• Create realistic, data-driven financial projections.

In our specific case, Apple's revenue grown from \( 3.68 \) billion to \( 15.68 \) billion over five years showcases the practical application of such a model. By determining the growth rate, stakeholders can better envision the trajectory of financial progress and allocate resources effectively.