An engineer is designing a precision component using a series of circular discs of decreasing size, arranged in a straight line so that each disc is tangential to the next.
The area of each disc is exactly 40% of the area of the previous disc in the sequence. The radius of the first (largest) disc is R R\,R mm.
Find, in terms of RRR, the radius of the second disc in the sequence.
Assuming an infinite number of discs are placed in this manner, show that the total horizontal length of the sequence of discs will always be less than 5.5R5.5R5.5R.
The design is refined to include a fixed gap of 5 mm between the edges of every adjacent pair of discs. Explain how this refinement affects the algebraic model for the total length, and state why the total length would no longer approach a finite limit as the number of discs increases.
