Optimization of Fencing Cost for a Garden Adjacent to a Road.As a property owner, you want to fence a garden which is adjacent to a road. The fencing next to the road must be stronger and cost $6 per foot. The fencing on the other sides cost $4 per foot. The area of garden is 2400 square feet.

QUESTION

As a property owner, you want to fence a garden which is adjacent to a road. The fencing next to the road must be stronger and cost $6 per foot. The fencing on the other sides cost $4 per foot. The area of garden is 2400 square feet.

Draw several diagrams to express the situation and calculate the cost for each configuration, and then estimate the dimension of minimum cost.
Find the function that represents the cost in terms of one of its sides.
Using the graphing utility, graph this cost function
Find the dimension that minimizes the cost of fencing and compare this with your estimate based the diagram you made.
Requirements:

Your paper should be 2-3 pages in length (not counting the title page and references page) and cite and integrate at least one credible outside source. The CSU-Global Library (Links to an external site.) is a great place to find resources. Your textbook is a credible resource.
Include a title page, introduction, body, conclusion, and a reference page.
The introduction should describe or summarize the topic or problem. It might discuss the general applications of the topic or it might introduce the unique terminology associated with the topic.
The body of your paper should address the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
The conclusion should summarize your thoughts about what you have determined from your analysis in completing the assignment. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
Include any tables of data or calculations, calculated values, and/or graphs referenced in the paper. (Note: The minimum required length excludes any tables, graphs, etc.)
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ANSWER

Optimization of Fencing Cost for a Garden Adjacent to a Road

Introduction

As a property owner, the task is to fence a garden that is adjacent to a road. The goal is to determine the most cost-effective configuration by considering different fencing costs for each side. This paper will explore various configurations, calculate the cost for each, and estimate the dimensions that minimize the cost of fencing.

Approach and Calculation

To analyze the cost of different configurations, let’s consider the garden as a rectangular shape with one side adjacent to the road. We will label this side as ‘x’ and the other two sides as ‘y.’ The area of the garden is given as 2400 square feet, which can be expressed as xy = 2400.

Configuration 1: Fencing on the adjacent side (road side)

In this configuration, the adjacent side requires a stronger fence costing $6 per foot, while the other two sides cost $4 per foot. The cost equation for this configuration is C = 6x + 4(2x + y).

Configuration 2: Fencing on opposite sides (non-road sides)
In this configuration, the non-road sides require the stronger fence, while the adjacent side costs $4 per foot. The cost equation for this configuration is C = 4y + 4(2y + x).

Configuration 3: Fencing on all sides

In this configuration, all sides require the stronger fence, resulting in a cost equation of C = 6x + 4(2x + y) + 4y.

Graphing the Cost Function

To graph the cost function, we can choose one of the configurations. Let’s select Configuration 1. We will plot the cost (C) on the y-axis and the length of the adjacent side (x) on the x-axis. The graph will provide a visual representation of how the cost changes with different dimensions.

Analysis and Dimension of Minimum Cost

By evaluating the cost equations for each configuration and analyzing their graphs, we can determine the dimensions that minimize the cost of fencing. The minimum cost corresponds to the lowest point on the graph.

Conclusion

In conclusion, this analysis demonstrates the importance of considering different fencing configurations and their associated costs when deciding how to fence a garden adjacent to a road. By calculating and graphing the cost function for various configurations, we can identify the dimensions that result in the minimum cost. This information enables property owners to make informed decisions and optimize their fencing expenses while meeting their needs.