I agree, though I prefer replacing x - 100 by x + 60. This stresses that the sine wave begins at x = -60 and reaches its first minimum at x = 60. However, replacing x - 100 by x + 60 does not change the function itself. Let's denote this function by w(x).

First draw a graph. We see that summer takes place during the second dip. Moreover, w(171) and w(265) are greater than 220 (and even greater than the average weight 227.5), so the interval when Michael’s weight is below 220 during the second dip is entirely inside summer. This means that we just need to find the length of this interval (and do not subsequently need to take intersection with the summer interval).

Since (220 - 227.5) / 22.5 = 1/3, we need to find the length of the interval inside the second dip when w(x) deviates more than 1/3 of its amplitude from the average 227.5. This is the same as the length of the interval inside the first wave of the function w'(x) = sin(2pi/160*x) where w'(x) > 1/3. Solve 2pi / 160 * x = sin^{-1}(1/3). Then the length in question is semi-period minus 2 * x.