Problem 17 - Applied Calculus

Question

Annual revenue \(R\) from McDonald's restaurants world. wide can be estimated by \(R=19.1+1.8 t,\) where \(R\) is in billion dollars and \(t\) is in years since January \(1,2005 .^{13}\) (a) What is the slope of this function? Include units. Interpret the slope in terms of McDonald's revenue. (b) What is the vertical intercept of this function? Include units. Interpret the vertical intercept in terms of McDonald's revenue. (c) What annual revenue does the function predict for \(2015 ?\) (d) When is annual revenue predicted to hit 35 billion dollars?

Step-by-Step Solution

Answer

(a) Slope: 1.8 billion dollars/year; revenue increases annually by 1.8 billion. (b) Vertical intercept: 19.1 billion dollars; revenue at start of 2005. (c) Revenue for 2015: 37.1 billion dollars (d) Revenue hits 35 billion in late 2013.

Step 1: Identify the equation

The given equation for McDonald's revenue is \( R = 19.1 + 1.8t \) where \( R \) is the revenue in billion dollars and \( t \) is the number of years since January 1, 2005.

Step 2: Calculate and interpret the slope

The slope of the function is 1.8. This indicates that McDonald's annual revenue is expected to increase by 1.8 billion dollars for each additional year since 2005. The units of the slope are billion dollars per year.

Step 3: Determine the vertical intercept

The vertical intercept is 19.1 billion dollars, which represents the estimated revenue at the start of 2005 (\( t = 0 \)). This means that McDonald's annual revenue at the beginning of 2005 was approximately 19.1 billion dollars.

Step 4: Predict revenue for 2015

To predict the revenue for 2015, calculate \( t = 2015 - 2005 = 10 \). Substitute \( t = 10 \) into the equation: \( R = 19.1 + 1.8 \times 10 \). This gives \( R = 19.1 + 18 = 37.1 \). The predicted revenue for 2015 is 37.1 billion dollars.

Step 5: Determine when revenue hits 35 billion dollars

Set \( R = 35 \) and solve for \( t \):\( 35 = 19.1 + 1.8t \)Subtract 19.1 from both sides:\( 15.9 = 1.8t \)Divide both sides by 1.8:\( t = \frac{15.9}{1.8} \approx 8.83 \).Since \( t = 8.83 \) represents the number of years since 2005, adding it to 2005 gives approximately 2013.83, which is around October 2013.

Key Concepts

Slope Interpretation

The slope of a linear function is a crucial concept, as it represents how much one variable changes concerning another. In the case of the McDonald’s revenue function, the equation provided is \( R = 19.1 + 1.8t \). Here, the slope is 1.8, which specifically indicates the rate of change of the revenue \( R \) with respect to time \( t \).

This means:

• Each year, McDonald’s revenue is expected to increase by 1.8 billion dollars.
• The slope provides insight into the growth of revenue over time, helping stakeholders understand financial progression.
• The units of the slope are "billion dollars per year," which underscores its practical interpretation in financial terms.

Understanding the slope helps in predicting future business prospects and aligning strategic goals accordingly.

Revenue Prediction

Predicting revenue is a key application of linear functions in business. For McDonald's, the equation \( R = 19.1 + 1.8t \) can be used to predict future revenues for specific years.

For instance, determining the annual revenue forecast for 2015 involves the following steps:

• Calculate \( t \) for the year 2015: \( t = 2015 - 2005 = 10 \).
• Substitute \( t = 10 \) into the revenue equation.
• Solve to find \( R = 19.1 + 1.8 \times 10 = 37.1 \).

The predicted revenue for 2015 is 37.1 billion dollars.

This prediction aids in planning, budgeting, and decision-making, as it allows businesses to prepare for future financial situations.

Vertical Intercept

The vertical intercept in the context of linear functions is the point where the graph intersects the vertical axis (also known as the \( y \)-axis). In the revenue function for McDonald's, the intercept is 19.1 billion dollars.

This specific value holds an essential interpretation:

• It represents the initial revenue at the starting point, which is January 1, 2005 (\( t = 0 \)).
• The intercept provides a baseline for analyzing changes over time, offering a reference point for all future growth.
• This number helps understand the company's financial standing before any subsequent growth or decline due to years passing.

Overall, recognizing the vertical intercept can help stakeholders create a clear financial history and track growth effectively.