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# minimum distance

Go to solution Solved by Yoruichi-san,

## Question

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. What is the minimum value of L?

My image won't upload so you can see it here:

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• Solution

The height of point P, i.e. the distance between P and D.

The total distance of rope or whatever was used is, as a function of the height of P, which is denoted x:

f(x)=sqrt(x^2+4)+sqrt(x^2+9)+(5-x)

Taking the derivative is pretty simple:

f'(x)=x/sqrt(x^2+4)+x/sqrt(x^2+9)-1

Set f'(x)=0 to find the minimum, and after a giant mess of algebra (which apparently I suck at :/) or plugging it into a numerical solver , you'll get a root of x=1.407.

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For those who can't access this image:

There is a triangle BPC where on one line segment BC there is a point D such that BD=2 and DC=3. Extending through P is a line segment going from some point A to D which is a distance of 5 and perpendicular to BC. Find the minimum total distance of all the "cables" by placing p in an optimal place.

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The point is placed ~0.099917m or about 10cm from the bottom. Too lazy to write up the equations, basically used calculus to find the minimum (can never get enough calculus ;P)

The height of A really doesn't matter, since that term disappears when you take the derivative, which makes sense, if you think about it.

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Thanks, kukupai.

I haven't fully worked out the equations, but I don't think that's it.

Using DP = 0.1 m, I get

AP = 4.9 m
BP > 2 m
CP > 3 m
Sum > 9.9 m
Using DP = 1 m, I get
AP = 4 m
BP = sqrt(5) m
CP = sqrt(10) m
Sum < 9.4 m
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Avoiding tortuous trig derivatives:

Put P at (2, y).

1. sin B + sin C = 1 (Condition for extremum is dy/dBP + dy/dCP = dy/dy)
2. tan B / tan C = 3/2 (Inspection)

Guess B reasonably, say 45o.

Solve  for C. Check .

Increment B by small amount and repeat.

Update B based on change.

Repeat a few times.

B = 35.12633457oC = 25.12653497o

y = 2 tan B = 1.406997295

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I got the same figure.

But, with plasmid, I don't know where the cables are.

Assumption (from post 1): We are to minimize PA + PB + P

yes. PA + PB + PC = L

Actually you're right, the answer is 1.407, apparently my algebra with square roots sucks ;P

I am not sure what 1.407 is referring to but i get a different answer.

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The height of point P, i.e. the distance between P and D.

The total distance of rope or whatever was used is, as a function of the height of P, which is denoted x:

f(x)=sqrt(x^2+4)+sqrt(x^2+9)+(5-x)

Taking the derivative is pretty simple:

f'(x)=x/sqrt(x^2+4)+x/sqrt(x^2+9)-1

Set f'(x)=0 to find the minimum, and after a giant mess of algebra (which apparently I suck at :/) or plugging it into a numerical solver , you'll get a root of x=1.407.

I see the issue, I was looking for L the total minimal length of PA,PB,PC while you found the length of DP. Nice job.

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