Answer to Question #22863 in Math for Rob Johnson

Question

Hi, I'm doing a project for college involving essentially a huge torsion spring loaded tape measure.

I've been struggling with this problem for a while and no matter how much i research I cannot find a solution! Basically, I need to work out the reel diameter if all of my tape is retracted fully.

The tape is 50mm wide and 1mm thick and 120ft long.
Any help would be much appreciated!

Answer 1

1ft = 30,48cm = 304,8mm;
120ft=36576mm;
When we rotate the tape length of first circle is PI anddiameter equals, length of second circle is 2PI and diameter equals 2, etc.;
In this way in every circle we spend d*PI mm of tape, so ifwe have diameter d=n, we already have spent PI(1+2+...+n)mm of tape.
In this way PI(1+2+...+n)<=36756.
PI*(1+n)*n/2<=36756;
(1+n)*n<=2*36756/PI;
n^2+n-73512/PI<=0;
73512/PI=23400;
If we solve this problem, we get that -153<=n<=152;but n is natural, that's why 0<n<=152; max{n}=152; In this way answer:
diameter equals 152mm.