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MarkFL
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Here is the question:
I have posted a link there to this thread so the OP can view my work.
I have posted a link there to this thread so the OP can view my work.
Firedawn's questions at Yahoo Answers regarding minimizing cost of pipeline
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In summary, the optimal location for laying the pipeline to minimize the cost is approximately 5.1 km from point A.
- #2
MarkFL
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Hello Firedawn,
Let's first draw a diagram of the path of the pipeline. All distances are in kilometers.
View attachment 1754
The powerhouse is at $\text{P}$, and the island is at $\text{I}$. The path of the pipeline is drawn in red. Let $C$ be the cost to lay the pipeline over land. The total cost is the cost per unit length time the total length, hence we may express the total cost as a function of $x$ as follows:
\(\displaystyle C(x)=C(13-x)+\frac{7}{5}C\sqrt{x^2+5^2}\)
Differentiating with respect to $x$ and equating the result to zero, we obtain:
\(\displaystyle C'(x)=-C+\frac{7}{5}C\frac{x}{\sqrt{x^2+5^2}}=0\)
Multiply through by \(\displaystyle \frac{5\sqrt{x^2+5^2}}{C}\)
\(\displaystyle -5\sqrt{x^2+5^2}+7x=0\)
\(\displaystyle 7x=5\sqrt{x^2+5^2}\)
Square both sides:
\(\displaystyle 49x^2=25x^2+625\)
\(\displaystyle x^2=\frac{625}{24}\)
Take the positive root:
\(\displaystyle x=\frac{25}{12}\sqrt{6}\approx5.10310363079829\)
To determine the nature of the extremum associated with this critical value, we may use the second derivative test:
\(\displaystyle C'(x)=-C+\frac{7}{5}C\frac{x}{\sqrt{x^2+5^2}}=0\)
\(\displaystyle C''(x)=0+\frac{7}{5}C\frac{\sqrt{x^2+5^2}(1)-x\left(\dfrac{x}{\sqrt{x^2+5^2}} \right)}{\left(\sqrt{x^2+5^2} \right)^2}=\frac{35C}{\left(x^2+5^2 \right)^{\frac{3}{2}}}\)
We see that for all real $x$ the second derivative is positive, hence our critical value is at the global minimum.
Let's first draw a diagram of the path of the pipeline. All distances are in kilometers.
View attachment 1754
The powerhouse is at $\text{P}$, and the island is at $\text{I}$. The path of the pipeline is drawn in red. Let $C$ be the cost to lay the pipeline over land. The total cost is the cost per unit length time the total length, hence we may express the total cost as a function of $x$ as follows:
\(\displaystyle C(x)=C(13-x)+\frac{7}{5}C\sqrt{x^2+5^2}\)
Differentiating with respect to $x$ and equating the result to zero, we obtain:
\(\displaystyle C'(x)=-C+\frac{7}{5}C\frac{x}{\sqrt{x^2+5^2}}=0\)
Multiply through by \(\displaystyle \frac{5\sqrt{x^2+5^2}}{C}\)
\(\displaystyle -5\sqrt{x^2+5^2}+7x=0\)
\(\displaystyle 7x=5\sqrt{x^2+5^2}\)
Square both sides:
\(\displaystyle 49x^2=25x^2+625\)
\(\displaystyle x^2=\frac{625}{24}\)
Take the positive root:
\(\displaystyle x=\frac{25}{12}\sqrt{6}\approx5.10310363079829\)
To determine the nature of the extremum associated with this critical value, we may use the second derivative test:
\(\displaystyle C'(x)=-C+\frac{7}{5}C\frac{x}{\sqrt{x^2+5^2}}=0\)
\(\displaystyle C''(x)=0+\frac{7}{5}C\frac{\sqrt{x^2+5^2}(1)-x\left(\dfrac{x}{\sqrt{x^2+5^2}} \right)}{\left(\sqrt{x^2+5^2} \right)^2}=\frac{35C}{\left(x^2+5^2 \right)^{\frac{3}{2}}}\)
We see that for all real $x$ the second derivative is positive, hence our critical value is at the global minimum.
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Related to Firedawn's questions at Yahoo Answers regarding minimizing cost of pipeline
1. How can I minimize the cost of building a pipeline?
There are several ways to minimize the cost of building a pipeline. One approach is to conduct a thorough cost analysis to identify potential areas for cost savings, such as using alternative materials or adjusting the route of the pipeline. Another way is to negotiate with suppliers to obtain lower prices for materials and equipment. Additionally, implementing efficient construction practices and closely monitoring expenses can also help reduce costs.
2. What are some common cost-saving strategies for pipeline construction?
Some common cost-saving strategies for pipeline construction include optimizing the route of the pipeline to minimize land acquisition and construction costs, using prefabricated components to reduce labor costs, and implementing environmentally friendly practices to avoid potential fines or delays. Additionally, collaborating with local communities and stakeholders can help minimize potential legal or regulatory costs.
3. Is it possible to minimize the cost of a pipeline without compromising on quality?
Yes, it is possible to minimize the cost of a pipeline without compromising on quality. This can be achieved by carefully selecting materials and construction methods that are both cost-effective and meet industry standards. It is also important to conduct regular inspections and maintenance to ensure the pipeline remains in good condition and prevent potential costly repairs in the future.
4. How can I estimate the cost of pipeline construction?
Estimating the cost of pipeline construction can be a complex process and may require the assistance of experts. However, some key factors to consider include the length and diameter of the pipeline, terrain and environmental conditions, labor and equipment costs, and potential regulatory fees. It is important to conduct a thorough analysis and consider all potential expenses to obtain an accurate cost estimate.
5. Are there any innovative technologies that can help minimize the cost of pipeline construction?
Yes, there are several innovative technologies that can help minimize the cost of pipeline construction. For example, using advanced mapping and surveying tools can help optimize the route of the pipeline and reduce the need for costly land acquisition. Additionally, implementing remote monitoring systems can help identify and address potential issues before they become major problems, saving both time and money in the long run.
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