Answer
The string stretches by $1.945m$.
Work Step by Step
1) The climber's velocity after he falls $0.75m$ is $$v_f^2=v_0^2+2gs$$ $$v_f^2=0+2(9.8)(0.75)=14.7$$ $$v_f=3.83m/s$$
2) Because energy is conserved, $$E_0=E_f (1)$$
We take the initial point to be when the rope runs out of slack, so $h_0=0$ and $y_0=0$. The climber's speed $v_0=3.83m/s$
The final point is when the rope stretches to the maximum of length $y_f (y_f\lt0$ because the climber falls). The climber's height then $h_f=y_f$. The climber's speed $v_f=0$
From (1), we have: $$mgy_f+\frac{1}{2}ky_f^2=\frac{1}{2}mv_0^2$$
The climber's weight $mg=842.8N$ and $k=1.2\times10^3N/m$. The initial KE $\frac{1}{2}mv_0^2=\frac{1}{2}(86)(3.83)^2=630.76J$. Therefore, $$(0.6\times10^3)y_f^2+842.8y_f-630.76=0$$
After solving the equation, because $y_f\lt0$, we get $y_f=-1.945m$
