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Visualizing An Infinity You Can't See But Know is There

Note: Sometimes old-but-not-olde math texts include some problems involving some interesting ideas. Pulled again from a text designed for engineering students, this week's problems deal with a visual infinity!

Note: Try to make a guess of the "totals" before working out the problems...

Problem 1: In a 60o right triangle (above), find the sum of all the perpendiculars to the base a, a1, a2, ...if the series is continued indefinitely?

Problem 2: The isosceles triangle (above) is completely filled with tangent circles as shown. Find the total area of all the circles.


Source: H. Rice & R. Knight's Technical Mathematics With Calculus, 1966

Hint: Work in sequence...find the first length (or area), the second length (or area), ...can you generalize what is happening?


Solution Commentary: No answers provided....rather, argue the merits of your solutions with someone else...