Answer
$16miles$ of the abandoned road.
It would cost more. That is $8,246,000$ dollars.
Work Step by Step
At first, we have to assign values or variables to the parts of the road that we will do calculates on (For a better visualization see the image above).
We will assign $x$ to the part of the abandoned road that will not be restored and is within the $40 mi$ range pointed on the image.
So, the restored road will be $(40-x)mi$.
And let's call the new road $n$.
Note, that Foxton is to the north of the abandoned road, so the line drawn on the image will be perpendicular to the abandoned road. As a graphical visualization of this image we can imagine a right-angle triangle (See the image above).
According to the Pythagoras theorem, we can calculate $n$:
$n=\sqrt{x^2+10^2}$
For the part of the restored road the cost will be:
$100,000(40-x)\$$
And for the part of new road the cost will be:
$200,000(\sqrt{x^2+100})\$$
To fit in the budget of $6,800,000\$$, the total cost should be equal to this amount:
$100,000(40-x) + 200,000(\sqrt{x^2+100}) = 6,800,000$ Divide by $100,000$
$40-x+2\sqrt{x^2+100}=68$
$2\sqrt{x^2+100}=x+28$
$4(x^2+100)=(x+28)^2$
$4x^2+400=x^2+56x+784$
$3x^2-56x-384=0$
$D=b^2-4ac=3136+4608=7744$
$x_1=\frac{-b-\sqrt{D}}{2a}=\frac{56-88}{6}=-\frac{16}{3}$ The distance cannot be negative, so we cross out this result.
$x_2=\frac{-b+\sqrt{D}}{2a}=\frac{56+88}{6}=\frac{144}{6}=24$
We have $x=24$ which is unrestored part of the road.
Part of the abandoned road to use is $40-24=\underline{16}$
If they directly built a new road between these cities, the distance would be $a$, which is hypotenuse of a triangle with sides of $40$ and $10$, so to calculate cost of it (which is $200,000\$$ per mile, as they have to build the whole new road) we can write (According to the Pythagoras theorem):
$a=\sqrt{40^2+10^2} = \sqrt{1600+100}=\sqrt{1700}\approx41.23mi$
In total it would cost: $41.23\times 200,000=8,246,000\$$
As we can clearly see it costs much more.
