Least Expensive Circuit

Suppose we have an ideal \(5V\) DC voltage source and three load resistors \((R_1,R_2,R_3) = (1 \Omega, 3 \Omega, 10 \Omega)\).

We are tasked with connecting the source to the load resistors using a wire topology consisting of one straight trunk and three straight branches (shown in red in the diagram). The trunk/branch junction is located somewhere in the 2D plane. \(xy\) coordinates of the source and load connection points are also shown in the diagram.

The red wire has a resistance of \(0.5 \Omega\) per unit length.

There are penalties for using a lot of wire, and penalties for not delivering rated voltage to the load resistors (\(5V\)). These penalties are quantified using a cost function:

\[C = 4 \, L + (v_1 - 5)^2 + (v_2 - 5)^2 + (v_3 - 5)^2\]

In the cost function, \(L\) is the total length of red wire, and \(v_1,v_2,v_3\) are the voltages across the load resistors.

What is the minimum possible value of the cost function (to one decimal place)?

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