1. **State the problem:** We are given the equation for total deformation $\delta$ as the sum of three parts: $$\delta = \delta_{AB} + \delta_{BC} + \delta_d$$ with the numerical equation: $$8 \times 10^{-3} = \frac{(100 \times 10^3)(250 \times 10^{-3})}{(4 \times 10^9) E \pi \left(\frac{0.1}{2}\right)^2} + 0 + \frac{(100 \times 10^3)(250 \times 10^{-3})}{(4 \times 10^9) \pi \left(0.1 - \frac{d}{2}\right)^2}$$ We need to solve for $d$. 2. **Understand the formula:** The deformation $\delta$ is given by the sum of deformations in different segments. Each deformation term is calculated as: $$\delta = \frac{Force \times Length}{E \times Area}$$ where $E$ is Young's modulus, and $Area = \pi r^2$ for circular cross-sections. 3. **Simplify known terms:** - Force $F = 100 \times 10^3$ - Length $L = 250 \times 10^{-3}$ - Young's modulus $E = 4 \times 10^9$ - Radius for first term $r_1 = \frac{0.1}{2} = 0.05$ Calculate the first term: $$\delta_{AB} = \frac{(100 \times 10^3)(250 \times 10^{-3})}{(4 \times 10^9) \pi (0.05)^2}$$ Calculate numerator: $$100 \times 10^3 \times 250 \times 10^{-3} = 100000 \times 0.25 = 25000$$ Calculate denominator: $$4 \times 10^9 \times \pi \times (0.05)^2 = 4 \times 10^9 \times \pi \times 0.0025 = 10^7 \pi$$ So: $$\delta_{AB} = \frac{25000}{10^7 \pi} = \frac{25000}{31415926.54} \approx 7.96 \times 10^{-4}$$ 4. **Rewrite the equation:** $$8 \times 10^{-3} = 7.96 \times 10^{-4} + \frac{25000}{(4 \times 10^9) \pi \left(0.1 - \frac{d}{2}\right)^2}$$ Subtract $7.96 \times 10^{-4}$ from both sides: $$8 \times 10^{-3} - 7.96 \times 10^{-4} = \frac{25000}{(4 \times 10^9) \pi \left(0.1 - \frac{d}{2}\right)^2}$$ Calculate left side: $$8 \times 10^{-3} - 7.96 \times 10^{-4} = 0.008 - 0.000796 = 0.007204$$ 5. **Solve for $\left(0.1 - \frac{d}{2}\right)^2$:** $$0.007204 = \frac{25000}{(4 \times 10^9) \pi \left(0.1 - \frac{d}{2}\right)^2}$$ Invert both sides: $$\left(0.1 - \frac{d}{2}\right)^2 = \frac{25000}{(4 \times 10^9) \pi \times 0.007204}$$ Calculate denominator: $$4 \times 10^9 \times \pi \times 0.007204 \approx 4 \times 10^9 \times 3.1416 \times 0.007204 = 9.05 \times 10^7$$ Calculate right side: $$\frac{25000}{9.05 \times 10^7} \approx 2.76 \times 10^{-4}$$ 6. **Take square root:** $$0.1 - \frac{d}{2} = \sqrt{2.76 \times 10^{-4}} = 0.0166$$ 7. **Solve for $d$:** $$\frac{d}{2} = 0.1 - 0.0166 = 0.0834$$ $$d = 2 \times 0.0834 = 0.1668$$ **Final answer:** $$d \approx 0.167$$ meters