# Problem 607

**Marsh Crossing**

Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The marsh is 50 leagues wide at all points, and the mid-point of AB is located in the middle of the marsh. A map of the region is shown in the diagram below:

The marsh consists of 5 distinct regions, each 10 leagues across, as shown by the shading in the map. The strip closest to point A is relatively light marsh, and can be crossed at a speed of 9 leagues per day. However, each strip becomes progressively harder to navigate, the speeds going down to 8, 7, 6 and finally 5 leagues per day for the final region of marsh, before it ends and the terrain becomes easier again, with the speed going back to 10 leagues per day.

If Frodo and Sam were to head directly East for point B, they would travel exactly 100 leagues, and the journey would take approximately 13.4738 days. However, this time can be shortened if they deviate from the direct path.

Find the shortest possible time required to travel from point A to B, and give your answer in days, rounded to 10 decimal places.

**穿越沼泽**

弗罗多和山姆需要从点A向东移动100里格（长度单位）到达点B。在普通的土地上，他们每天可以走10里格，所以这趟旅程将会花费10天时间。但是，他们在路上需要穿过一个从正西南向正东北延伸的长沼泽，而他们在沼泽中行进时需要降低速度。沼泽在任意一处的宽度都是50里格，而AB连线的中点也恰好落在沼泽的中点。下图展示的是这片区域的地图：

沼泽由5个不同的条状部分组成，每个部分宽10里格，在地图中表现为深浅的不同。最靠近A的条状部分沼泽化程度最浅，在这部分的行进速度可以达到每天9里格。然而，随着向东行进，每个条状部分逐渐难以前行，行进速度依次降低为每天8、7、6里格，直到最后一部分只有每天5里格；随后沼泽部分结束，回到正常的土地上，行进速度恢复到每天10里格。

如果弗罗多和山姆径直前往点B，他们恰好移动100里格，而这趟旅程将会花费约13.4738天。然而，如果他们不走直路，他们可以缩短旅行所需的时间。

找出从点A到点B所需的最少旅行时间，以天为单位，并保留小数点后10位小数。