Problem 28 - Calculus and its applications

Question

Intel, a computer chip manufacturer, reported \(\$ 1265\) million in total revenue in \(1986 .\) In \(2012,\) the total revenue was \(\$ 53.3\) billion. Assuming an exponential model, find the growth rate \(k\), to four decimal places, and determine the revenue function \(R\), with \(R(t)\) in billions of dollars. Then predict the company's total revenue for 2020.

Step-by-Step Solution

Answer

The growth rate \( k \) is approximately 0.1323. The revenue function is \( R(t) = 1.265 e^{0.1323t} \). Predicted revenue for 2020 is approximately $318.931 billion.

Step 1: Understand Exponential Growth

We assume revenue grows exponentially, represented by the equation \( R(t) = R_0 e^{kt} \). Here, \( R_0 \) is the initial revenue, \( k \) is the growth rate, and \( t \) is the time in years since the initial measurement.

Step 2: Identify Known Values and Time Intervals

Given: \( R_0 = 1.265 \) billion dollars in 1986, \( R(t) = 53.3 \) billion dollars in 2012. The time interval \( t \) from 1986 to 2012 is 26 years.

Step 3: Set Up Exponential Growth Equation

Using the formula \( R(t) = R_0 e^{kt} \), plug in the known values to get: \ \( 53.3 = 1.265 e^{26k} \).

Step 4: Solve for Growth Rate \( k \)

To isolate \( k \), first divide both sides by \( 1.265 \): \ \( e^{26k} = \frac{53.3}{1.265} \) \ Next, calculate the division: \ \( e^{26k} \approx 42.1307 \) \ Take the natural logarithm of both sides: \ \( 26k = \ ln(42.1307) \) \ Solve for \( k \): \ \( k \approx \frac{ln(42.1307)}{26} \approx 0.1323 \) as the growth rate to four decimal places.

Step 5: Define the Revenue Function \( R(t) \)

Substitute \( R_0 = 1.265 \) and \( k = 0.1323 \) into the exponential equation: \ \( R(t) = 1.265 e^{0.1323t} \).

Step 6: Predict Revenue for 2020

Calculate \( t \) for the year 2020: \( 2020 - 1986 = 34 \) years. Substitute \( t = 34 \) into the revenue function: \ \( R(34) = 1.265 e^{0.1323 \times 34} \). \ Compute the result: \ \( R(34) \approx 1.265 e^{4.4922} \approx 318.931 \) billion dollars.

Key Concepts

Growth Rate Calculation

Calculating the growth rate is crucial to understanding how a quantity expands over time. In the case of Intel's revenue, we approached this through the concept of exponential growth.

Exponential growth is when something increases at a rate proportional to its current value. The formula we use is: \[ R(t) = R_0 e^{kt} \] where:

• \( R(t) \) is the quantity at time \( t \)
• \( R_0 \) is the initial quantity
• \( k \) is the growth rate
• \( e \) is the base of the natural logarithm (approximately 2.71828)

In the exercise, Intel's initial revenue was \( 1.265 \) billion dollars, and by 2012 it had grown to \( 53.3 \) billion dollars. To find the growth rate \( k \), we rearrange the equation and solve for \( k \). By using natural logarithms to isolate \( k \), we arrive at a growth rate of approximately 0.1323, which is a crucial step for further predictions.

Revenue Prediction

Predicting future revenue requires an understanding of how the growth model projects numbers into the future. With the growth rate \( k \) identified, we use the exponential growth equation:

\[ R(t) = 1.265 e^{0.1323t} \]This formula allows us to predict Intel's revenue for any given year by substituting the number of years since 1986 into \( t \).

For example, to find the revenue for 2020:- First, calculate \( t \) as the number of years from 1986 to 2020, which is 34.- Then plug \( t \) into the revenue function: \[ R(34) = 1.265 e^{0.1323 \times 34} \]By computing this expression, we predict that Intel's revenue for 2020 will be approximately \( 318.931 \) billion dollars. This prediction relies on the assumption that the exponential growth continues at the same rate.

Exponential Functions

Exponential functions are a cornerstone in modeling growth processes where the rate of change is proportional to the current value. This makes them ideal for representing scenarios like population growth, radioactive decay, and financial calculations, such as predicting a company's revenue.

The general form of an exponential function is: \[ f(x) = a e^{bx} \]where:

• \( a \) is the initial value or starting point
• \( b \) represents the rate of growth or decay
• \( x \) is the independent variable, often representing time

In the context of predicting revenue, the exponential function allows us to model how revenue grows when reinvested earnings compound over time. It is especially powerful because it reflects the reality that as the revenue expands, the absolute increase accelerates due to ongoing reinvestment of profits.

Understanding exponential functions provides insight into how and why certain entities grow rapidly. It is essential for interpreting long-term predictions and making informed decisions based on historical trends.