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Minimize Sum of Line Segments Length w/ Point P on Line AD - Yahoo Answers
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In summary, the question is asking to find the minimum value of the total length of cables connecting a point P to points A, B, and C on the line AD. This can be represented by the function L(x) = x + √(x²-8x+17) + √(x²-8x+25), where x is the distance between P and A. By differentiating and finding the root of the derivative, we estimate the minimum value of L to be approximately 7.58 meters.
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MarkFL
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Hello realsf,
First, we may define:
\(\displaystyle L=\overline{AP}+\overline{BP}+\overline{CP}\)
Next, let's label the diagram with the given information:
View attachment 1719
We are given:
\(\displaystyle \overline{AP}=x\)
And by Pythagoras, we find:
\(\displaystyle \overline{BP}=\sqrt{1^2+(4-x)^2}=\sqrt{x^2-8x+17}\)
\(\displaystyle \overline{CP}=\sqrt{3^2+(4-x)^2}=\sqrt{x^2-8x+25}\)
Hence, we may give $L$ as a function of $x$ as follows:
\(\displaystyle L(x)=x+\sqrt{x^2-8x+17}+\sqrt{x^2-8x+25}\)
Differentiating with respect to $x$, we obtain:
\(\displaystyle \frac{dL}{dx}=1+\frac{2x-8}{2\sqrt{x^2-8x+17}}+\frac{2x-8}{2\sqrt{x^2-8x+25}}=1+\frac{x-4}{\sqrt{x^2-8x+17}}+\frac{x-4}{\sqrt{x^2-8x+25}}\)
Observing we must have $0\le x\le 4$, here are the plots of $L(x)$ and \(\displaystyle \frac{dL}{dx}\) on the relevant domain:
View attachment 1720
Using a numeric root finding technique intrinsic to the CAS, we obtain:
View attachment 1721
Hence, we find:
\(\displaystyle L_{\min}\approx7.58\)
First, we may define:
\(\displaystyle L=\overline{AP}+\overline{BP}+\overline{CP}\)
Next, let's label the diagram with the given information:
View attachment 1719
We are given:
\(\displaystyle \overline{AP}=x\)
And by Pythagoras, we find:
\(\displaystyle \overline{BP}=\sqrt{1^2+(4-x)^2}=\sqrt{x^2-8x+17}\)
\(\displaystyle \overline{CP}=\sqrt{3^2+(4-x)^2}=\sqrt{x^2-8x+25}\)
Hence, we may give $L$ as a function of $x$ as follows:
\(\displaystyle L(x)=x+\sqrt{x^2-8x+17}+\sqrt{x^2-8x+25}\)
Differentiating with respect to $x$, we obtain:
\(\displaystyle \frac{dL}{dx}=1+\frac{2x-8}{2\sqrt{x^2-8x+17}}+\frac{2x-8}{2\sqrt{x^2-8x+25}}=1+\frac{x-4}{\sqrt{x^2-8x+17}}+\frac{x-4}{\sqrt{x^2-8x+25}}\)
Observing we must have $0\le x\le 4$, here are the plots of $L(x)$ and \(\displaystyle \frac{dL}{dx}\) on the relevant domain:
View attachment 1720
Using a numeric root finding technique intrinsic to the CAS, we obtain:
View attachment 1721
Hence, we find:
\(\displaystyle L_{\min}\approx7.58\)
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FAQ: Minimize Sum of Line Segments Length w/ Point P on Line AD - Yahoo Answers
What is the purpose of minimizing the sum of line segments length with point P on line AD?
The purpose of minimizing the sum of line segments length with point P on line AD is to find the shortest possible path that connects all the given points on the line AD. This optimization problem is often encountered in various fields such as engineering, mathematics, and computer science.
How is the length of line segments calculated in this problem?
In this problem, the length of a line segment is calculated using the distance formula, which is the square root of the sum of the squared differences between the x and y coordinates of the endpoints of the line segment. This formula can be applied to each line segment in the path to determine the total length.
What is the role of point P in this problem?
Point P serves as a reference point or a pivot for the line segments. It is a fixed point on the line AD and the remaining points are connected to this point to create a path. By minimizing the sum of line segments length with point P on line AD, we are essentially finding the optimal placement of point P that results in the shortest path.
Can this problem be solved using mathematical equations?
Yes, this problem can be solved using mathematical equations and optimization techniques such as calculus, linear programming, and dynamic programming. These methods can help find the value of point P that minimizes the sum of line segments length on line AD.
Are there any real-world applications of this problem?
Yes, this problem has many real-world applications such as finding the shortest route for delivery trucks, minimizing travel time for public transportation, and optimizing the placement of cell phone towers for maximum coverage. It can also be applied in circuit design and layout optimization in engineering.