MHB Minimize Total Length of Cables: Estimate Minimum Value?

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To minimize the total length of cables linking point P to points A, B, and C, the function L(x) is defined using the Pythagorean theorem. The derivative L'(x) is calculated and set to zero to find critical values, leading to the application of Newton's method for approximation. Starting with an initial guess, the iterations converge to a value of approximately 1.407 for x. The second derivative test confirms this is a global minimum, resulting in a minimum cable length of approximately 9.35 meters. The calculations demonstrate an effective method for estimating minimum values in optimization problems.
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Here is the question:

How to estimate the minimum value?

A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B, and C is minimized (see the figure). Estimate the minimum value of L to two decimals.

View attachment 2123

So far I have:

dL/dx=1 + ((x-5)/(sqrt(29-10 x+x^2))) + ((x-5)/(sqrt(50-10 x+x^2)))

What is the next step??

I have posted a link there to this thread so the OP can view my work.
 

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Hello Lavender V,

Let's let $0\le x\le 5$ represent the length of segment $\overline{DP}$ in meters. And so the total length of cable as a function of $x$ is then obtained via the Pythagorean theorem as:

$$L(x)=(5-x)+\sqrt{x^2+2^2}+\sqrt{x^2+3^2}$$

Now, we may differentiate with respect to $x$ and equate the result to zero to obtain the critical value(s):

$$L'(x)=-1+\frac{x}{\sqrt{x^2+4}}+\frac{x}{\sqrt{x^2+9}}=0$$

Now, since we are thankfully allowed to approximate, we may use Newton's method here to approximate the root of this equation.

Let:

$$f(x)=-1+\frac{x}{\sqrt{x^2+4}}+\frac{x}{\sqrt{x^2+9}}$$

We need to compute the first derivative, and so for the last two terms, consider:

$$g(x)=\frac{x}{\sqrt{x^2+k}}$$

Using the quotient, power and chain rules, we obtain:

$$g'(x)=\frac{\sqrt{x^2+k}(1)-x\dfrac{x}{\sqrt{x^2+k}}}{\left(\sqrt{x^2+k} \right)^2}=\frac{k}{\left(x^2+k \right)^{\frac{3}{2}}}$$

And so we may conclude that:

$$f'(x)=\frac{4}{\left(x^2+4 \right)^{\frac{3}{2}}}+\frac{9}{\left(x^2+9 \right)^{\frac{3}{2}}}$$

Newton's method gives us the recursion:

$$x_{n+1}=x_{n}-\frac{f\left(x_n \right)}{f'\left(x_n \right)}$$

And so using the definition of $f$ and its derivative, we have:

$$x_{n+1}=x_{n}-\frac{-1+\dfrac{x_n}{\sqrt{x_n^2+4}}+\dfrac{x_n}{\sqrt{x_n^2+9}}}{\dfrac{4}{\left(x_n^2+4 \right)^{\frac{3}{2}}}+\dfrac{9}{\left(x_n^2+9 \right)^{\frac{3}{2}}}}$$

Now, using a computer or CAS, and $x_0=1$, we find:

$$x_1\approx1.36825580018177$$

$$x_2\approx1.40658793851904$$

$$x_3\approx1.40699724822532$$

$$x_4\approx1.40699729457262$$

$$x_5\approx1.40699729457262$$

This is accurate to 15 digits, more than enough to get an approximation for the minimum of $L$ to two decimal places. Now, if we note that:

$$f'(x)=L''(x)>0$$

For all real values of $x$, then by the second derivative test, we may conclude that this critical value is at the global minimum, and we need not check the boundaries.

Thus, we may conclude:

$$L_{\min}\approx L(1.40699729457262)\approx9.35\text{ m}$$
 
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