I linked the towns in this kind of fashion
A.`````````````````````B
\ ____________ /
/``````````````````` \
B```````````````````````C
Ignore the `
Then I found the minimum if this function :
4(√(106.5^2+n^2))+(284-2n), 0≤n≤142
I get 21600 million dollars.
355^2 = 213^2 + 284^2
ABCD (labelled by beginning letters of each town) is a rectangle.
The shortest road would be a cross through the rectangle.. so 355 + 355 = 710 miles
OR
AD (213 miles) + DC (284 miles) + CB (213 miles)
or a similar variation
DA (213 miles) + AB (284 miles) + BC (213 miles)
In any case.. the combined length of roads (minimum) seems to be 710 miles.
I don’t see how it could be less than that.
road miles, $m$ ...
$m = 4\sqrt{x^2+106.5^2} + (284-2x)$
$\dfrac{dm}{dx} = \dfrac{4x}{\sqrt{x^2+106.5^2}} - 2$
$\dfrac{dm}{dx} = 0 \implies \dfrac{2x}{\sqrt{x^2+106.5^2}} = 1$
$4x^2 = x^2+106.5^2$
$x = \dfrac{106.5}{\sqrt{3}} \implies m \approx 653$ miles