I’m building a ladder that is 70 feet tall. The ladder decreases in size proportionally from the standard size ladder at the bottom at the bottom to zero at the top.The first rung is at 12 inches above the ground. I am looking for is a descendent ratio in inches, rung to rung.

If r is the ratio, 1/(1-r)=70, 1-r=1/70, so r=69/70.

The reason is that the length of the ladder is the sum of the series 1+r+r²+r³+... where r<1 where 1 is the distance between the ground and first rung (12 inches=1 foot). The second rung is 69/70 ft (about 11.83 inches) from the first rung, and the third rung is (69/70)² ft = 0.9716 ft (approx 11.66 inches) from the second rung, and so on. There are an infinite number of rungs. The sum to infinity is 1/(1-r).

Note that the ratio is a fraction, not the number of inches. If the distance between the rungs was decreased by a fixed number of inches it would not be a ratio. The graph below shows a fixed ratio of 69/70. The x-axis shows how many rungs would make the length of the ladder the value on the y-axis. The blue line is an asymptote at 70 feet, and an infinite number of rungs is needed to get to that length. As x gets bigger the length of the ladder gets closer to 70 feet. If a fixed decrement (not a ratio) of 12/139 of an inch is applied to each rung then there will be 139 or 140 rungs. 12/139" is the maximum fixed decrement.

by Top Rated User (695k points)