Fountain Fractal

One day, Jonas was randomly doodling in the margins of his notes during lecture for his introductory quantum mechanics course. He was drawing all kinds of lines, squiggles, circles, rectangles, and other outlines of shapes. One of these was a mere, isosceles triangle.

"Hmmm...How could I make this more interesting?" he thought. He drew a line from the center of the base (with length $2b$ ) to one of the equal sides, such that it is orthogonal to that side (i.e. forms a right angle). Symmetrical about the bisector of the vertex angle (the line which defines the triangle's height, $h$ ), he drew another orthogonal line to the other equal side. Afterwards, he connected the ends of these two lines with a horizontal line (orthogonal to the bisector line). From here, he continued this process iteratively off of each of these successive horizontal lines, generating a fractal comprised of similar isosceles triangles, drawn like the figure shown, except in black-and-white (given that he doodled in pencil).

Assume that Jonas is a very detailed doodler, and therefore drew this fractal to an infinite number of iterations. After completing his fractal, he looked up at the board and realized that he had spent so long drawing, he lost what the professor was talking about. Given that he drew at a constant rate of $1 { cm }/{ s }$ , calculate how long he spent on drawing the fractal (the grey part of the figure), in seconds.

Details and Assumptions

• $b=5cm$ (half of the triangle's base)
• $h=12cm$ (the triangle's height)