Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq 900 .\) Explain what the solution of the inequality represents.

Step-by-Step Solution

Answer

The Bed Bath & Beyond stores will reach or exceed 900 locations part-way through the year 2006.

Step 1: Setting Up the Inequality

Plug the value of \(B(t)\) from the given model into the inequality. This yields the inequality: \(86.5t + 342 \geq 900.\)

Step 2: Solving the Inequality

Solving this inequality is done by starting with subtracting 342 from both sides, resulting in \(86.5t \geq 558.\) Then, to isolate \(t\), divide both sides by 86.5, hence \( t \geq 6.45.\)

Step 3: Interpretation of the Solution

The variable \(t\) represents the number of years from the year 2000. So, \(t \geq 6.45\) implies that Bed Bath & Beyond will have 900 or more stores 6.45 years after 2000, which is part-way through the year 2006.

Key Concepts

Linear Equations

Linear equations are fundamental elements in mathematics, used commonly to represent relationships with a constant rate of change. A linear equation takes the form of \( ax + b = c \), where \( a \) and \( b \) are constants and \( x \) is the variable that we solve for. In the context of the exercise, the equation for Bed Bath & Beyond stores can be expressed as:
\[ B = 86.5t + 342 \]
This equation tells us that the number of stores, \( B \), grows linearly over time, \( t \).

• \( 86.5 \) is the rate of change or slope, indicating that approximately 87 stores are added each year.
• \( 342 \) is the y-intercept, representing the initial number of stores in the year 2000.

Solving linear equations often involves isolating the variable, like we see in the step-by-step solution: subtracting \( 342 \) from 900 and then dividing by 86.5. This straightforward approach highlights why linear equations are essential tools for predicting and managing growth over time.

Quadratic Equations

Quadratic equations appear when dealing with relationships where the rate of change is not constant. They are typically expressed in the form \( ax^2 + bx + c = 0 \). In the exercise, the equation for Williams-Sonoma stores is a quadratic model:
\[ W = -2.92t^2 + 52.0t + 381 \]
This specific equation illustrates a scenario where growth doesn't happen at a steady rate.

• The term \(-2.92t^2\) suggests a negative acceleration, indicating that at some point, the number of stores would start decreasing if extrapolated forward.
• The linear term \( 52.0t \) complements the quadratic term, adjusting the overall rate of growth.
• The constant \( 381 \) is the store count at the start, just like the initial value in a linear model.

Quadratic equations are more complex and versatile, ideal for modeling more intricate real-world scenarios where growth can peak and then decline.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures to represent real-world phenomena. In this exercise, mathematical models are created to predict the number of stores over time.
Using the equations given:

• Linear models are powerful for situations with steady changes, perfect for Bed Bath & Beyond which has consistent growth.
• Quadratic models suit dynamically changing scenarios found in Williams-Sonoma, where growth might accelerate or decelerate over time.

These models help businesses to plan strategically, such as determining when a store count will hit certain milestones, like Bed Bath & Beyond reaching 900 stores. By setting models into inequalities, predictions about reaching specific thresholds or limits become possible. This is a key application of mathematical modeling, making abstract concepts useful in real-life decisions.

Real-World Applications

In real-world applications, both linear and quadratic equations offer insight for decision-making across various fields. Businesses like retail chains use these mathematical tools to project growth and understand market trends.
For Bed Bath & Beyond, solving \( B(t) \geq 900 \) is a practical illustration of how a company can forecast achieving business goals. Being able to determine when the store count will exceed a certain number helps in setting strategic objectives like expansion plans and resource allocation.
Similarly, understanding the implications of a quadratic model for Williams-Sonoma can highlight times when growth will slow, allowing for financial adjustments.

• This type of predictive modeling is crucial for budgeting, marketing strategies, and staffing requirements.
• Insights drawn from these models guide leaders in making informed policy or investment choices.

Ultimately, these equations translate numerical data into actionable business insights, providing clear advantages for entities aiming to thrive in competitive environments.