Calculus and Geometry

[Monsta]

Please help me with this problem, I got a solution but I don't think it's correct.

A, B and C are computers. A Printer needs to be connected to each of them. Where does P need to be placed in order to minimize the total cable length AP + BP + CP?

Sketch:

]

The equation I got is:

Square root of (4² + P²) + Square root of (5²+ P²) + 8-P.

However, this does not seem to be correct and if I try to find the minimum I get strange values. Please tell me my mistake and help me. Thank you very much for your help.

This is another problem, and the coordinates confuse me. P has to be on the line and the aim is to find out where P has to be so that the sum of the length of the lines AP, BP and CP is as short as possible.

 

[galactus]

A, B and C are computers. A Printer needs to be connected to each of them. Where does P need to be placed in order to minimize the total cable length AP + BP + CP?

Sketch:

]

The equation I got is:

Square root of (4² + P²) + Square root of (5²+ P²) + 8-P.

However, this does not seem to be correct and if I try to find the minimum I get strange values. Please tell me my mistake and help me. Thank you very much for your help.

You should get something around 2.6?.

\(\displaystyle \L\\\frac{dD}{dp}=\frac{p}{\sqrt{p^{2}+25}}+\frac{p}{\sqrt{p^{2}+16}}-1=0\)

Solving this for p gives p=2.57797955202

Plugging this back into your formula gives a minimum of 15.806275

Though, it is handy to know that the max and mins occur at the same points for the distances and squares of the distances.
The above is a monster to solve for p. So, we can use the squares.

\(\displaystyle \L\\(p^{2}+16)+(p^{2}+25)+(8-p)^{2}=3p^{2}-16p+105\)

Differentiate and get \(\displaystyle \L\\6p-16\)

Set to 0 and solve for p and get \(\displaystyle \L\\p=\frac{8}{3}=2.666667\)

Which is very close to the first answer.