Problem 101
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 equation \(B(t)=W(t) .\) Explain what the solution of the equation represents.
Step-by-Step Solution
Verified Answer
The solution of \(t\) represents the year (from 2000) when Bed Bath & Beyond and William-Sonoma will have the same quantity of stores.
1Step 1: Set up the Equation to Solve
First set the two equations equal to each other to solve for \(t\), \(86.5t+342=-2.92t^2+52.0t+381\).
2Step 2: Simplify the Equation
To simplify the equation, put it in standard form, which results in a quadratic equation. This is achieved by moving the terms on the left-hand side of the equation to the right, yielding \(0= -2.92t^2 -34.5t +39\).
3Step 3: Solve the Quadratic Equation
The quadratic equation can be solved using the quadratic formula. This formula is \(t = [-b \pm sqrt(b^2 - 4ac)] / 2a\). Applying this to the quadratic equation, \(a = -2.92\), \(b = -34.5\) and \(c = 39\), yields \(t = [34.5 \pm sqrt((-34.5)^2 - 4 * -2.92 * 39)] / 2 * -2.92\). Solving this yields two potential solutions for \(t\).
4Step 4: Check the Solution
Check both solutions by inserting each one into the original equations, and ensure their results are equal. If they are, the solution is valid. If not, it is invalid. Hence, the valid solution will be the year when both companies will have the same number of stores.
Key Concepts
Mathematical ModelingAlgebraic ExpressionsReal-world Applications
Mathematical Modeling
Mathematical Modeling allows us to represent real-world situations in a mathematical framework. In this exercise, we use mathematical models to predict the number of stores for Bed Bath & Beyond and Williams-Sonoma over a period.
This involves constructing equations that approximate real-world data. For instance, the linear equation for Bed Bath & Beyond, \(B = 86.5t + 342\), and the quadratic equation for Williams-Sonoma, \(W = -2.92t^2 + 52.0t + 381\), each model how the number of stores changes over time.
These models allow us to analyze trends and make predictions for years beyond the given data. Through mathematical modeling, we can simulate conditions in the real world and solve problems involving growth or change in a systematic way.
This involves constructing equations that approximate real-world data. For instance, the linear equation for Bed Bath & Beyond, \(B = 86.5t + 342\), and the quadratic equation for Williams-Sonoma, \(W = -2.92t^2 + 52.0t + 381\), each model how the number of stores changes over time.
These models allow us to analyze trends and make predictions for years beyond the given data. Through mathematical modeling, we can simulate conditions in the real world and solve problems involving growth or change in a systematic way.
Algebraic Expressions
An Algebraic Expression is a mathematical phrase that can contain numbers, variables, and operations. In the given exercise, both store models are algebraic expressions.
Bed Bath & Beyond's expression is linear, \(B = 86.5t + 342\), which reflects a consistent growth of stores over time. The coefficient of \(t\), 86.5, indicates the rate of increase annually.
In contrast, the Williams-Sonoma model is quadratic, \(W = -2.92t^2 + 52.0t + 381\). Here, the \(-2.92t^2\) term causes the expression to reflect a parabolic structure. This indicates that growth may not be constant over time, suggesting a peak followed by decline.
Understanding these algebraic expressions is crucial for forming equations that can be solved to find intersections or points of equality between two different models.
Bed Bath & Beyond's expression is linear, \(B = 86.5t + 342\), which reflects a consistent growth of stores over time. The coefficient of \(t\), 86.5, indicates the rate of increase annually.
In contrast, the Williams-Sonoma model is quadratic, \(W = -2.92t^2 + 52.0t + 381\). Here, the \(-2.92t^2\) term causes the expression to reflect a parabolic structure. This indicates that growth may not be constant over time, suggesting a peak followed by decline.
Understanding these algebraic expressions is crucial for forming equations that can be solved to find intersections or points of equality between two different models.
Real-world Applications
Quadratic Equations often have Real-world Applications, such as determining when different business models reach similar benchmarks. In this case, the solution to the equation \(B(t) = W(t)\), when both companies have the same number of stores, is found using algebraic manipulation and the quadratic formula.
This represents the point in time (year) when Bed Bath & Beyond will have as many stores as Williams-Sonoma. This is valuable for business planning and competitive analysis, helping the companies strategize.
This represents the point in time (year) when Bed Bath & Beyond will have as many stores as Williams-Sonoma. This is valuable for business planning and competitive analysis, helping the companies strategize.
- Companies can assess whether to expand or cut down operations.
- It allows for resource allocation to gain an advantage.
- Provides a timeline for growth benchmarks against competitors.
Other exercises in this chapter
Problem 100
use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\)
View solution Problem 100
The total public debt \(D\) (in trillions of dollars) in the United States from 2005 through 2014 can be approximated by the model $$D=0.051 t^{2}+0.20 t+5.0,5
View solution Problem 101
The metabolic rate of an ectothermic organism increases with increasing temperature within a certain range. Experimental data for oxygen consumption \(C\) (in m
View solution Problem 102
use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\)
View solution