Surprisingly hard paper roll problem :)

[Guest raisinflames]

Hi everyone!

 

Recently, in our calculus class, we solved some related rates problems, and I decided to think of one myself. Then I figured out that its not as easy as it seemed! Maybe someone can give me a helping hand in it:

 

So, here's how it goes:

A roll of paper of thickness 0.1 mm (you can assume any numbers, really) is pulled at a constant rate of 20 m/min. How fast does it radius decrease around the radius of 5 cm (in other words, what is dr/dt?). I made several assumptions to simplify the problem, but they didn't help much: for example, the minimum radius is 0, and the paper is unrolled not spiral-like, but in concentric circles with increasing radiuses (sort of like Bohr's atom model

But, still, I ended up with part of my equation being (1+2+3+...+n), where n is the number of times the paper is rolled around, and is probably also a function of t. So, here is as far as I went... my teacher said the problem could be solved by mathematical induction (a topic we yet did not covered). If so, how?

 

 

P.S. the problem provided to be in a strange way similar to the "classic challenge" in a physic forum here.

[uncool]

Mathematical induction is a way of saying "If it works for some integer, and we know that if it works for one integer, it works for the integer after it, then it works for all integers." Here, mathematical induction can be used in the following manner:

 

Let f(n) = 1 + 2 + 3 + ... + n

I say that f(n) = n*(n+1)/2

 

It works for 1, because f(1) = 1 = 1*2/2.

f(n+1) = 1 + 2 + 3 + ... + n + n+1 = f(n) + n

So if f(n) = n*(n+1)/2, then f(n+1) = n*(n+1)/2+n+1 = (n+1)*(n/2 + 1) = (n+1)*(n+2)/2 = (n+1)*(n+1+1)/2.

Therefore, the formula is correct for all numbers past 1.

-Uncool-

[Dave]

I felt I needed to add something to the previous post. Let's assume we have some statement that we want to prove, and its dependent upon some number n - for example, your triangle number problem. We want to prove that 1 + ... + n = n(n+1)/2.

 

To prove something by induction, we need to do the following things:

1) Prove it for n = n₀, where n₀ is just some "starting" value.

2) Assuming that it's true for some other number k, prove it for k+1.

Then the statement is true for all numbers greater than or equal to n₀.

 

You can use the rest of uncool's proof to complete the rest.