Answer
$2.9\times10^{-2}m$
Work Step by Step
Let's apply Newton's second law to the skier,
$F=ma$
$F-f=ma$ ; Where F - towing force, f - resistive force due to water.
$F=f+ma=130\space N+(59\space kg)(0.85\space m/s^{2})=180.15\space N$
According to equation 10.17 the change $\Delta L$ in the length of the rope is,
$\Delta L=\frac{FL_{0}}{YA}$ ; Let's plug known values into this equation.
$\Delta L=\frac{180.15\space N(12\space m)}{(3.7\times10^{9}N/m^{2})(2\times10^{-5}m^{2})}\approx2.9\times10^{-2}m$
Here $Y=3.7\times10^{9}N/m^{2}$ for nylon is taken from table 10.1
