Alan

Like this:

Nicely done Melody.  Here are the answers found by Mathcad (I should have said 6, not 9):

Here's one way (messy though!):

arclength = radius * subtended angle  i.e. \(27.4\times 7\pi /4\)

You can crunch the numbers.

(8*3 + 2*9 + n*9)/(n + 10) = 8

You can find the number of slips, n, from this.

1.  Draw a graph of the left-hand side against w and the right-had side against w to see where they intersect (perhaps multiply both sides by cos(w) first so that you don't get any divide by zeros).

2. Use a numerical technique like Newton-Raphson (though it's still a good idea to draw a graph first).

See https://web2.0calc.com/questions/help-asap-also-provide-how-u-got-ur-answer#r1

Like so:

I'll leave you to do the rest.

This should make it clearer:

Hmm.  For the roots of the equation to be rational it requires more than just that the discriminant is greater than or equal to zero.  It also requires that the square root of the discriminant is not irrational.