In a right triangle, a series of altitudes are drawn starting with an altitude drawn using the vertex of the right angle and drawn towards the hypotenuse. Then subsequently continuing to draw altitudes from the right angles in the new right triangles created, and which also include the vertex from the smallest angle in the original right triangle. The series of altitudes are drawn so they move closer and closer to the smallest angle in the original right triangle. Find the sum of the lengths of these altitudes given that one of the angles of the original triangle is 55 degrees and the side of the triangle adjacent to the this angle is 22.4 meters long.