Problem 80 - Algebra A Combined Function

Question

The amount of money (in billions of dollars) spent by the Boeing Company and subsidiaries on research and development annually is represented by the function \(R(x)=0.54 x+1.96,\) where \(x\) is the number of years since 2004. (Source: Based on data from the Boeing Corporation) a. Find and interpret \(R(2)\). b. Estimate the amount of money spent on research and development by Boeing in \(2009 .\)

Step-by-Step Solution

Answer

R(2) is $3.04 billion, indicating 2006 R&D spending. Boeing spent $4.66 billion in 2009.

Step 1: Understand the Function

The function given is \(R(x) = 0.54x + 1.96\), where \(x\) is the number of years since 2004. This function tells us the amount in billions of dollars spent on R&D by Boeing each year.

Step 2: Calculate R(2)

To find \(R(2)\), substitute \(2\) for \(x\) in the function: \(R(2) = 0.54(2) + 1.96\). Calculate to find \(R(2) = 1.08 + 1.96 = 3.04\).

Step 3: Interpret R(2)

The result \(3.04\) means that in 2006 (which is 2 years after 2004), Boeing spent approximately $3.04 billion on R&D.

Step 4: Determine x for 2009

To find \(x\) for the year 2009, calculate the number of years since 2004: \(2009 - 2004 = 5\). Thus, \(x = 5\).

Step 5: Estimate R(5)

Substitute \(5\) for \(x\) in the function: \(R(5) = 0.54(5) + 1.96\). Calculate to find \(R(5) = 2.7 + 1.96 = 4.66\).

Step 6: Interpretation of R(5)

The value \(4.66\) indicates that in 2009, Boeing spent approximately $4.66 billion on research and development.

Key Concepts

Understanding Algebra in Linear Functions

Algebra is like a mathematical toolbox. It helps us solve problems using functions, equations, and expressions. When we talk about linear functions in algebra, we mean equations that form straight lines when graphed. These functions are usually written in the form of \(y = mx + b\).

In the equation, \(m\) represents the slope of the line, indicating how steep the line is. Meanwhile, \(b\) is the y-intercept, representing the value of \(y\) when \(x\) is 0. This framework allows us to model many real-world situations, like spending in research and development (R&D).

For instance, in Boeing's R&D spending function \(R(x) = 0.54x + 1.96\), \(0.54\) is the slope, showing how much the spending increases each year. \(1.96\) is the initial spend in 2004. This clear algebraic relationship gives us insights into how spending evolves over time.

Interpreting Function Values

Understanding what the values in a function represent is crucial. This process is called function interpretation. Let's take a deeper look into this.

When we substitute a value for \(x\) in a function like \(R(x) = 0.54x + 1.96\), we calculate the corresponding \(R(x)\). For example, \(R(2) = 0.54(2) + 1.96 = 3.04\) tells us Boeing's R&D spending in 2006 was \(3.04 billion. By finding \(R(5) = 4.66\), we know it increases to \)4.66 billion in 2009.

• This calculated \(R(x)\) represents actual or projected spending at specific times.
• Using the function, we can see how factors change with time.

Thus, interpreting the values helps us understand not only the present scenario but also helps predict future trends.

Creating Mathematical Models

Mathematical modeling involves using mathematical structures to represent real-world systems. It's like creating a simple version of a complex situation. In the case of Boeing's spending, we use the function \(R(x) = 0.54x + 1.96\) to model the costs related to R&D over different years.

This linear model helps:

• Predict future spending by extrapolating from past data.
• Analyze trends such as constant growth rate given by the slope \(0.54\).
• Plan budgets by projecting future values.

The key to successful modeling is creating equations that accurately capture important aspects of the situation being studied. This makes it easier for stakeholders to make informed decisions based on the model's predictions.

Research and Development Insights

Research and development (R&D) plays a critical role in innovation and long-term growth. Boeing’s function \(R(x) = 0.54x + 1.96\) provides insight into the company's commitment to R&D over time.

The positive slope \(0.54\) indicates a steady annual increase in R&D spending. This suggests Boeing’s priority in advancing technology and staying competitive. As of 2009, their increase reveals accumulation in investment over each year.

• A firm’s R&D spending is a key indicator of their focus on innovation.
• Function interpretation allows industry analysts to gauge how much emphasis a company places on development.

Such assessments are beneficial not just to the company but also to investors and policy makers monitoring industry trends.