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...