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Chapter 4: Problem 28

(Compare with the preceding exercise.) You are thinking of making your mansion more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs \(\$ 4\) and saves you an average of \(\$ 2\) per year in energy costs, and each square foot of wall insulation costs \(\$ 1\) and saves you an average of \(\$ 0.20\) per year in energy costs. \(^{13}\) Your mansion has 200 light fittings and 3,000 sq. ft. of uninsulated exterior wall. To impress your friends, you would like to spend as much as possible, but save no more than \(\$ 800\) per year in energy costs (you are proud of your large utility bills). How many compact fluorescent light bulbs and how many square feet of insulation should you purchase? How much will you save in energy costs per year?

Short Answer

Expert verified
In order to spend as much as possible while still saving no more than \(\$800\) per year in energy costs, you can purchase 200 compact fluorescent light bulbs and 2,000 square feet of insulation. This will cost you \(\$3800\) and will result in annual energy cost savings of \(\$1000\). However, this exceeds the desired \(\$800\) per year limit. To meet the limit exactly, you may need to adjust the amount of insulation or light bulbs slightly.

Step by step solution

01

Define the variables

Let: - x = the number of compact fluorescent light bulbs to be purchased - y = the amount of wall insulation in square feet to be purchased
02

Constraints and Objective Function

According to the problem, we have the following constraints: 1. Annual energy cost savings cannot exceed \(\$800\). Therefore, \(2x + 0.20y \le 800\) 2. The mansion has 200 light fittings. So the number of light bulbs cannot exceed 200. \(x \le 200\) 3. The mansion has 3,000 sq. ft. of uninsulated exterior wall. So the amount of wall insulation cannot exceed 3,000 sq. ft. \(y \le 3000\) 4. The number of light bulbs and the amount of wall insulation cannot be negative. \(x \ge 0\) \(y \ge 0\) The objective is to spend as much as possible, given by the equation: \(C = 4x + y\)
03

Form a Linear Programming Problem

Our linear programming problem is: Maximize: \(C = 4x + y\) Subject to the constraints: 1. \(2x + 0.20y \le 800\) 2. \(x \le 200\) 3. \(y \le 3000\) 4. \(x \ge 0\) 5. \(y \ge 0\)
04

Solve the Problem using the Method of Vertices

We need to find the feasible region satisfying the constraints. The feasible region is a polygon with vertices. We will check the objective function at each vertex and look for the maximum value of C. Vertices: 1. Intersection of \(x=0, y=3000\) 2. Intersection of \(x=200, y=3000\) 3. Intersection of \(x=200, 2x + 0.20y = 800\) 4. Intersection of \(x=0, 2x + 0.20y = 800\) Calculating the coordinates of the vertices: 1. Vertex 1: (0,3000) 2. Vertex 2: (200,3000) 3. Vertex 3: Solving the system of equations: \(x = 200\) and \(2x + 0.20y = 800\): \[2(200) + 0.20y = 800\] \[0.20y = 400\] \[y = 2000\] Thus, Vertex 3: (200,2000) 4. Vertex 4: Solving the system of equations: \(x = 0\) and \(2x + 0.20y = 800\): \[0.20y = 800\] \[y = 4000\] Vertex 4 is not within the constraint \(y\le3000\), so we don't consider this vertex. Compute C at each of the three vertices: 1. C(Vertex 1) = 4(0) + 3000 = 3000 2. C(Vertex 2) = 4(200) + 3000 = 3800 3. C(Vertex 3) = 4(200) + 2000 = 2800 Since we want to maximize the cost, Vertex 2 provides the maximum cost, i.e., buying 200 light bulbs and 3,000 square feet of insulation.
05

Calculate the Annual Energy Cost Savings

To find the annual energy cost savings, we need to evaluate the expression \(2x + 0.20y\), where x = 200 and y = 3,000. \(2(200) + 0.20(3000) = 400 + 600 = 1000\) The annual energy cost savings will be \(\$1000\). To summarize, you should purchase 200 compact fluorescent light bulbs and 3,000 square feet of insulation to spend as much as possible. However, this would save you \(\$1000\) per year, which is more than the desired \(\$800\) per year limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Efficiency
Making a mansion more energy efficient involves changes that reduce energy consumption while maintaining comfort. These changes include switching to compact fluorescent light bulbs (CFLs) and adding wall insulation.
This specific exercise demonstrates how adding CFLs and wall insulation reduces annual energy costs. Each CFL contributes a savings of $2 per year, while each square foot of insulation saves $0.20 per year.
Understanding these savings helps in deciding the extent of the energy-efficient investments. By replacing regular bulbs with CFLs and insulating walls, significant cost reductions in energy bills can be achieved, enhancing environmental sustainability.
  • Compact Fluorescent Light Bulbs: Cost $4 each and save $2 annually in energy costs per bulb.
  • Wall Insulation: Costs $1 per square foot and saves $0.20 annually in energy costs per square foot.
Although installing CFLs and insulation can initially seem expensive, the long-term savings on utility bills not only lead to better energy efficiency but also contribute to a green lifestyle.
Constraints and Objective Function
In linear programming, constraints are conditions that must be met for a solution to be feasible. For this exercise:
  • The total annual energy savings must not exceed $800.
  • Maximum of 200 light fittings for CFLs.
  • Up to 3,000 square feet of wall can be insulated.
These constraints are represented mathematically as equations or inequalities. In this problem, let \( x \) be the number of CFLs and \( y \) be the amount of insulation in square feet.
We have:- Energy savings constraint: \( 2x + 0.20y \le 800 \)- Light bulbs limit: \( x \le 200 \)- Insulation limit: \( y \le 3000 \)- Non-negativity clauses: \( x \ge 0 \), \( y \ge 0 \)
The objective function expresses what needs to be maximized or minimized. Here, it's to maximize the spending cost given by \( C = 4x + y \).
This objective function, along with the constraints, forms the core of the linear programming model. Ideally, it guides the decision on the number of light bulbs and insulation to purchase while maximizing spending and respecting the constraints.
Feasible Region
The feasible region is a visual representation of all possible solutions that meet the constraints of a linear programming problem. It is typically bounded by lines which represent the constraints. In this example, the region is defined on a coordinate plane where x represents light bulbs and y represents insulation.
The vertices of the feasible region are essential because the optimal solution usually occurs at one of these points. Here, the constraints intersect and form vertices at several points.
  • Vertex 1: \((0, 3000)\)
  • Vertex 2: \((200, 3000)\)
  • Vertex 3: \((200, 2000)\)
The objective is to evaluate the objective function \( C = 4x + y \) at each vertex to find which maximizes spending.
Out of these, Vertex 2 provides the maximum spending value, suggesting purchasing 200 light bulbs and 3,000 square feet of insulation.
This assessment not only aids in grasping how constraints shape decision-making but also illustrates how linear programming graphs offer strategic insights into achieving optimal solutions.

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