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MarkFL
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Here is the question:
I have posted a link there to this topic so the OP can see my work.
I have posted a link there to this topic so the OP can see my work.
Lila Bird's question at Yahoo Answers regarding minimizing plot of land
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In summary, to find the dimensions of the rectangular plot with the smallest area that can be used for a swimming pool with a surrounding walkway of varying widths, one can use the general formula A(x,y) = xy, where A_P represents the area of the pool itself. Using the first derivative, the critical value for the horizontal length can be found, and using the second derivative, it can be determined that the extremum is a minimum. Plugging in the given data, the dimensions of the smallest plot are (15,10) in yards.
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MarkFL
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Hello Lila Bird,
I like to work such problems in general terms, which allows us to derive a formula that we can use for similar cases, and also to see how the various parameters affect the solution. So let's define:
\(\displaystyle A_P\) = the area of the pool itself.
\(\displaystyle w_x\) = the width of the walkway at the deep/shallow ends of the pool.
\(\displaystyle w_y\) = width of the walkway along the sides of the pool.
\(\displaystyle A\) = the area of the rectangular plot of land containing the pool and the surrounding walkway.
\(\displaystyle x\) = horizontal length of plot.
\(\displaystyle y\) = vertical length of plot.
Please refer to the following diagram:
View attachment 993
Thus, we may express the area of the plot as:
\(\displaystyle A(x,y)=xy\)
where we are constrained by:
\(\displaystyle A_P=\left(x-2w_x \right)\left(y-2w_y \right)\,\therefore\,y=\frac{A_P}{x-2w_x}+2w_y\)
And so we obtain the area of the plot in one variable $x$:
\(\displaystyle A(x)=x\left(\frac{A_P}{x-2w_x}+2w_y \right)\)
So, next we want to equate the first derivative to zero to find the critical value(s):
\(\displaystyle A'(x)=x\left(-\frac{A_P}{\left(x-2w_x \right)^2} \right)+(1)\left(\frac{A_P}{x-2w_x}+2w_y \right)=\frac{2\left(w_y\left(x-2w_x \right)^2-w_xA_P \right)}{\left(x-2w_x \right)^2}=0\)
Hence, this implies:
\(\displaystyle w_y\left(x-2w_x \right)^2-w_xA_P=0\)
Solving for $x$, and taking the positive root, we find the critical value:
\(\displaystyle x=\sqrt{\frac{w_x}{w_y}A_P}+2w_x\)
To determine the nature of the extremum associated with this critical value, we may use the second derivative test. We find:
\(\displaystyle A''(x)=\frac{4w_xA_P}{\left(x-2w_x \right)^3}\)
We can easily see that:
\(\displaystyle A''\left(\sqrt{\frac{w_x}{w_y}A_P}+2w_x \right)>0\)
Hence, the extremum is a minimum. Next we can find $y$ as follows:
\(\displaystyle y=\frac{A_P}{\sqrt{\frac{w_x}{w_y}A_P}}+2w_y=\sqrt{\frac{w_y}{w_x}A_P}+2w_y\)
Thus, we find the dimensions minimizing the plot of land subject to the constraint on the area of the pool are:
\(\displaystyle (x,y)=\left(\sqrt{\frac{w_x}{w_y}A_P}+2w_x, \sqrt{\frac{w_y}{w_x}A_P}+2w_y \right)\)
Now, to answer the specific problem given, we may plug in the data (in yards):
\(\displaystyle w_x=3,\,w_y=2,\,A_P=54\)
and we find:
\(\displaystyle (x,y)=(15,10)\)
I like to work such problems in general terms, which allows us to derive a formula that we can use for similar cases, and also to see how the various parameters affect the solution. So let's define:
\(\displaystyle A_P\) = the area of the pool itself.
\(\displaystyle w_x\) = the width of the walkway at the deep/shallow ends of the pool.
\(\displaystyle w_y\) = width of the walkway along the sides of the pool.
\(\displaystyle A\) = the area of the rectangular plot of land containing the pool and the surrounding walkway.
\(\displaystyle x\) = horizontal length of plot.
\(\displaystyle y\) = vertical length of plot.
Please refer to the following diagram:
View attachment 993
Thus, we may express the area of the plot as:
\(\displaystyle A(x,y)=xy\)
where we are constrained by:
\(\displaystyle A_P=\left(x-2w_x \right)\left(y-2w_y \right)\,\therefore\,y=\frac{A_P}{x-2w_x}+2w_y\)
And so we obtain the area of the plot in one variable $x$:
\(\displaystyle A(x)=x\left(\frac{A_P}{x-2w_x}+2w_y \right)\)
So, next we want to equate the first derivative to zero to find the critical value(s):
\(\displaystyle A'(x)=x\left(-\frac{A_P}{\left(x-2w_x \right)^2} \right)+(1)\left(\frac{A_P}{x-2w_x}+2w_y \right)=\frac{2\left(w_y\left(x-2w_x \right)^2-w_xA_P \right)}{\left(x-2w_x \right)^2}=0\)
Hence, this implies:
\(\displaystyle w_y\left(x-2w_x \right)^2-w_xA_P=0\)
Solving for $x$, and taking the positive root, we find the critical value:
\(\displaystyle x=\sqrt{\frac{w_x}{w_y}A_P}+2w_x\)
To determine the nature of the extremum associated with this critical value, we may use the second derivative test. We find:
\(\displaystyle A''(x)=\frac{4w_xA_P}{\left(x-2w_x \right)^3}\)
We can easily see that:
\(\displaystyle A''\left(\sqrt{\frac{w_x}{w_y}A_P}+2w_x \right)>0\)
Hence, the extremum is a minimum. Next we can find $y$ as follows:
\(\displaystyle y=\frac{A_P}{\sqrt{\frac{w_x}{w_y}A_P}}+2w_y=\sqrt{\frac{w_y}{w_x}A_P}+2w_y\)
Thus, we find the dimensions minimizing the plot of land subject to the constraint on the area of the pool are:
\(\displaystyle (x,y)=\left(\sqrt{\frac{w_x}{w_y}A_P}+2w_x, \sqrt{\frac{w_y}{w_x}A_P}+2w_y \right)\)
Now, to answer the specific problem given, we may plug in the data (in yards):
\(\displaystyle w_x=3,\,w_y=2,\,A_P=54\)
and we find:
\(\displaystyle (x,y)=(15,10)\)
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Related to Lila Bird's question at Yahoo Answers regarding minimizing plot of land
1. What is the significance of minimizing plot of land in scientific research?
Minimizing plot of land is important in scientific research because it allows for efficient use of resources, reduces environmental impact, and can lead to more accurate and reliable results.
2. How can we minimize plot of land in a research project?
One way to minimize plot of land in a research project is by using statistical techniques to ensure that the sample size is appropriate for the study and can provide enough data for meaningful analysis.
3. Are there any negative consequences of minimizing plot of land in a study?
While minimizing plot of land can have many benefits, it can also lead to a decrease in the representativeness of the study and may limit the generalizability of the results. It is important to carefully consider the trade-offs and potential limitations when minimizing plot of land in a study.
4. What factors should be considered when determining the appropriate plot of land for a study?
The appropriate plot of land for a study will depend on various factors, such as the research question, the type of data being collected, and the resources available. Other considerations may include the population size, sampling methods, and the level of precision needed for the study.
5. Can you provide an example of a study where minimizing plot of land was crucial to the results?
One example of a study where minimizing plot of land was crucial is in ecological research, where scientists often need to conduct studies on limited areas of land to minimize disturbance to the natural environment. By carefully selecting and minimizing plot of land, researchers can gather valuable data while minimizing their impact on the ecosystem.
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