Out of the woods

The man is at O.Draw a circle with centre O and radius 1.The edge of the woods is tangent to this circle.We are looking for the shortest curve which starts at O and has a common point with every tangent of the circle.I have decided to approach the problem in 4 stages.

Stage 1: Walk in a straight line in any direction for 1 mile to a point A.Then walk along the circumference of the circle.So I have to walk atmost $1+2π ≈ 7.28$ miles.(FIG 1)

Stage 2: But if we go through the path OABC, it also has a common point with every other tangent of the circle.So it leads out of the woods and its length is merely $3π/2+2 ≈ 6.71$ . miles (FIG 2)

Third Stage: The path OABCD also has a common point with every other tangent of the circle.Hence it will lead out of the woods in atmost 2+√2+π ≈ 6.556 miles. (FIG 3)

Fourth Stage: The path OABCD has the length $p(\alpha ,\beta )=|OA|+|AB|+arcBC+|CD|$ But $|OA|=(\cfrac { 1 }{ cos\quad\alpha } )$ , $|AB|=tan\alpha$ , $arcBC=2π-2\alpha -2\beta$ , $|CD|=tan\quad \beta$ , $\alpha$ and $\beta$ measured in radians.

$p(\alpha ,\beta )=2π+(\cfrac { 1 }{ cos\quad \alpha } +tan\quad \alpha -2\alpha )+(tan\quad \beta -2\beta )$ or $p(\alpha ,\beta )=2π+f(\alpha )+f(\beta ).$ . To minimize $p(\alpha ,\beta )$ we should minimize $f(\alpha)$ and $f(\beta)$ separately. But $f\acute { (\alpha ) } =\cfrac { (2sin\quad \alpha -1)(1+sin\quad \alpha ) }{ \cos ^{ 2 }{ \alpha } }$ and $g\acute { (\beta ) } =tan^{ 2 }\beta -1=(tan\beta -1)(tan\beta +1)$ . Since $\alpha$ and $\beta$ both are acute angles, $f\acute { (\alpha )=0 }$ and $g\acute { (\beta ) } =0$ and the unique solutions are $\alpha =\cfrac { π }{ 6 }$ and $\beta =\cfrac { π }{ 4 }$ . At these points, the signs of $f\acute { (\alpha ) }$ and $f\acute { (\beta ) }$ are changing from negative to positive.Thus we have minima at these values of the angles. The minimal path length is $p(\cfrac { π }{ 6 } ,\cfrac { π }{ 4 } )=1+\sqrt { 3 } +\cfrac { 7π }{ 6 } \approx$ 6.397. Similarly, it can be shown that there is no shortest path other than this.