Problem 77 - A Graphical Approach to College Algebra

Question

Sales of Apple Products Average household spending on Apple products is shown in the figure for both U.S. sales and worldwide sales. Use this figure (Figure can't copy) U.S. sales in dollars can be approximated during year \(x\) by $$ U(x)=13(x-2006)^{2}+115 $$ Evaluate \(U(2011)\) and interpret your result.

Step-by-Step Solution

Answer

U(2011) is $440, representing U.S. household spending on Apple products in 2011.

Step 1: Understand the Problem

The problem provides a formula for U.S. sales of Apple products in a given year, using the function \( U(x) = 13(x-2006)^{2} + 115 \). You need to calculate and interpret the sales for the year 2011.

Step 2: Substitute the Value into the Formula

Substitute \( x = 2011 \) into the equation \( U(x) = 13(x-2006)^{2} + 115 \). This will give us the U.S. sales for that particular year in the context of the provided model.

Step 3: Calculate the Expression Inside the Parentheses

Substitute \( x = 2011 \) and calculate the expression inside the parentheses: \( 2011 - 2006 \). This results in \( 5 \).

Step 4: Square the Result

Now, square the result from the previous step: \( 5^2 = 25 \).

Step 5: Multiply by 13

Multiply the squared result by 13: \( 13 \times 25 = 325 \).

Step 6: Add 115 to the Product

Add 115 to the product from the previous step to get the final result: \( 325 + 115 = 440 \).

Step 7: Interpretation of the Result

The result \( U(2011) = 440 \) represents the average household spending on Apple products in the U.S. in 2011, according to the model, which is $440.

Key Concepts

Problem Solving

When faced with a mathematical problem, the goal is to find an answer by using logical reasoning, formulas, and procedures. In this particular exercise, we are tasked with finding the U.S. sales of Apple products for the year 2011.
To tackle this problem, we begin by understanding the provided function, which is a mathematical representation of sales over time:

• Identify the year to evaluate, which is 2011 in this case.
• Substitute the year into the sales function to calculate the sales value.

These steps are essential to efficiently solve quadratic problems similar to this one. Each calculation builds on the previous, linking together like puzzle pieces to reveal the final answer. By breaking down each phase, problem-solving becomes systematic and intuitive.

Algebraic Interpretation

Algebraic interpretation involves understanding what each component of an equation represents and how it contributes to the final result. For this exercise, the primary function is: \[ U(x) = 13(x - 2006)^2 + 115 \] Here is how we can interpret this function in the context of the problem:

• The term \( (x - 2006) \) represents the number of years since 2006.
• Squaring this term \((x - 2006)^2\) emphasizes changes over time, indicating that differences grow quadratically.
• The coefficient 13 shows how much influence these changes in time have on sales.
• The constant term 115 is an initial baseline value of sales, independent of time.

By interpreting each component, we are better equipped to understand how the function derives the sales for 2011, providing insights into the mathematical relationship between time and sales growth.

Polynomials

A polynomial is an algebraic expression involving sums and powers of variables. The expression \( U(x) = 13(x - 2006)^2 + 115 \) is a quadratic polynomial, as it involves a squared term.Highlighted features of this polynomial include:

• It is a quadratic polynomial, since the highest power of \(x\) after expansion would be 2.
• The coefficients determine the shape and position of the graph it forms when plotted.

Quadratic polynomials often form parabolas, which in this scenario model how sales change over time from the base year of 2006. They provide a powerful tool in approximative real-world behaviors, like sales trends, through mathematical modeling.Understanding the nature of polynomials is central to solving algebraic problems, as they represent a broad class of functions with numerous applications in mathematics and beyond.