A ladder has rungs 25cm apart.The rungs decrease uniformly in length from 45cm at the bottom to 25cm at the top. If the top and button rungs are 2.5m apart.

Question

What is the length of the wood required for the rungs?

Answer 1

If the distance between the top and bottom rungs is 2.5m and rungs are 25cm apart, then you need 2.5/0.25 + 1 = 11 rungs for the ladder (you always need an extra one to fully enclose both ends). 

 

If the rungs are uniformly decreasing in length from 45cm to 25cm and there are 11 rungs, that means the rung lengths are: 45, 43, 41, 39, 37,....25. This is so because there are 45-25+1 = 21 numbers from 25 to 45 inclusively. And since the endpoints of the interval are both odd, there is n+1 odd numbers and n even numbers on that interval where 2n+1 = 21. Thus, there are 11 odd numbers and 10 even numbers and the rung lengths correspond to odd numbers. 

 

So now we have an arithmetic series that sums all odd numbers from 25 to 45 inclusive. The formula for that sum is (25+45) x n/2 where n is the number of terms in the series or, in this case, the number of rungs. 

 

(25+45) x (11/2) = 70 x 5.5 = 385cm

 

So you need 3.85m of wood for all the rungs.