Answer
The length of the rough patch is 1.28 meters.
Work Step by Step
$W = K_2-K_1$
$-F_f~d = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$
$-0.25~mg~d = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$
$-0.25~g~d = \frac{1}{2}v_2^2 - \frac{1}{2}v_1^2$
$d = \frac{v_2^2 - v_1^2}{(2)(-0.25~g)}$
$d = \frac{(1.65~m/s)^2 - (3.0~m/s)^2}{(2)(-0.25)(9.80~m/s^2)}$
$d = 1.28~m$
The length of the rough patch is 1.28 meters.
