# Answer to Question #22863 in Other Math for Rob Johnson

Question #22863

Hi, I'm doing a project for college involving essentially a huge torsion spring loaded tape measure.<br><br>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.<br><br>The tape is 50mm wide and 1mm thick and 120ft long.<br>Any help would be much appreciated!

Expert's answer

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.

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.

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