Problem 48 - A Graphical Approach to College Algebra

Question

The revenue in millions of dollars for the first 5 years of Internet advertising is given by \(A(x)=25(2.95)^{x},\) where \(x\) is years after the industry started. (Source: Business Insider.) (a) What was the Internet advertising revenue after 5 years? (b) Determine analytically when revenue was about \(\$ 250\) million. (c) According to this model, when did the Internet advertising revenue reach \(\$ 1\) billion?

Step-by-Step Solution

Answer

(a) \$5134.25 million after 5 years. (b) Around 1.35 years for \$250 million. (c) Reach \$1 billion after 2.75 years.

Step 1: Substitute x=5 into A(x) for Part (a)

To find the revenue after 5 years, substitute \(x=5\) into the function: \[ A(5) = 25 \times (2.95)^5 \]. Evaluate the expression to get the result.

Step 2: Calculate A(5) for Part (a)

Simplify \((2.95)^5\) to get approximately 205.37. Then, multiply 205.37 by 25: \[ A(5) = 25 \times 205.37 = 5134.25 \]Thus, the revenue after 5 years is roughly \$5134.25 million.

Step 3: Set up equation for Part (b)

To find when revenue was about \$250 million, set the function equal to 250 and solve for \(x\): \[ 25(2.95)^x = 250 \]. Divide both sides by 25 to simplify the equation.

Step 4: Solve for x using logarithms for Part (b)

Divide both sides by 25 to isolate the exponential term:\[ (2.95)^x = 10 \]. Take the natural logarithm of both sides to solve for \(x\):\[ \ln((2.95)^x) = \ln(10) \]. Use the property \(\ln(a^b) = b\ln(a)\):\[ x \ln(2.95) = \ln(10) \]. Divide both sides by \(\ln(2.95)\) to find \(x\).

Step 5: Calculate x for Part (b)

Calculate the final value:\[ x = \frac{\ln(10)}{\ln(2.95)} \approx 1.35 \].So, the revenue was about \$250 million after approximately 1.35 years.

Step 6: Set up equation for Part (c)

To find when revenue reached \\(1 billion, set \(A(x) = 1000\) (since \\)1 billion equals 1000 million): \[ 25(2.95)^x = 1000 \]. Divide both sides by 25 to begin simplifying.

Step 7: Solve for x using logarithms for Part (c)

After simplifying the equation we get:\[ (2.95)^x = 40 \]. Take the natural logarithm of both sides: \[ \ln((2.95)^x) = \ln(40) \]. Use the property \(\ln(a^b) = b\ln(a)\):\[ x \ln(2.95) = \ln(40) \]. Divide both sides by \(\ln(2.95)\) to solve for \(x\).

Step 8: Calculate x for Part (c)

Calculate the final value:\[ x = \frac{\ln(40)}{\ln(2.95)} \approx 2.75 \].So, the Internet advertising revenue reached \$1 billion after about 2.75 years.

Key Concepts

Understanding Internet Advertising Revenue

Internet advertising revenue has grown exponentially over time. In this exercise, we have a model represented by the equation \( A(x) = 25(2.95)^{x} \), where \( A(x) \) is the revenue in millions of dollars and \( x \) is the number of years after the industry began. This model allows us to forecast how revenues change over time by evaluating the growth factor, here \( 2.95 \).

• The base number, \( 2.95 \), signifies how many times the revenue grows each year.
• The initial factor of 25 reflects the starting revenue in millions of dollars.

By substituting different values for \( x \), we can determine revenue for specific future years. Such exponential growth models are vital in economics and business predictions and help stakeholders make informed decisions.

Solving with Logarithmic Equations

When revenues reach a particular milestone, logarithmic equations help find the required time period. To solve the model for a specific revenue, like \( \$250 \) million, we use logarithms. By transforming the equation \( 25(2.95)^{x} = 250 \) into logarithmic form, we can solve for \( x \).

Taking the natural logarithm on both sides and using the power rule \( \ln(a^b) = b \ln(a) \), simplifies solving the equation:

• Isolate the exponential expression: \( (2.95)^{x} = 10 \).
• Apply the logarithm: \( x = \frac{\ln(10)}{\ln(2.95)} \).

Thus, logarithms convert exponential equations into more manageable linear forms, making it simpler to calculate the duration until desired revenue levels are reached. These equations are powerful tools in not just business, but also in scientific and statistical computations.

Using Algebraic Modeling for Predictions

Algebraic modeling in real-world scenarios, like predicting internet advertising revenue, helps make complex decision-making more manageable. The model \( A(x) = 25(2.95)^{x} \) provides straightforward insights into how advertising budgets may perform over time.

• This model allows for quick calculations of revenue at any future point, yielding precise forecasts.
• It aids businesses in setting realistic goals based on past and projected growth rates.

By modeling such economic scenarios algebraically, we can experiment with variables and scenarios. For instance, changing the growth rate to test different market conditions. This predictive power is fundamental in strategic planning, enabling businesses to optimize their advertising efforts and maximize returns effectively.