# 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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