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.
ago by Level 1 User (120 points)

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

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.

ago by Top Rated User (651k points)

Related questions

1 answer
1 answer
asked Oct 18, 2011 in Word Problem Answers by anonymous | 184 views
1 answer
asked Apr 12, 2018 in Calculus Answers by anonymous | 46 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
83,025 questions
87,735 answers
1,969 comments
4,543 users