A steel cable 75 m long hangs straight down a deep well, holding nothing but itself…

Step 1. (a) Stress at the top $= \dfrac{\rho ALg}{A} = \rho gL = 7800\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2 \times 75\ \text{m} = 5.73 \times 10^{6}\ \text{Pa}$.

Step 2. (b) $\Delta L = \dfrac{\rho gL^2}{2Y} = 7800\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2 \times \dfrac{5625\ \text{m}^2}{4.0 \times 10^{11}\ \text{Pa}} = 1.08 \times 10^{-3}\ \text{m}$.

Step 3. (c) It breaks when $\rho gL = 4.5 \times 10^{8}\ \text{Pa}$: $L = \dfrac{4.5 \times 10^{8}\ \text{Pa}}{7800\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2} = 5887\ \text{m}$.