1. **Problem statement:** A copper cylinder is initially at 20.0°C. We want to find the temperature at which its volume is 0.150% larger than at 20.0°C. 2. **Formula and concepts:** The volume expansion of a solid due to temperature change is given by: $$\frac{\Delta V}{V_0} = \beta \Delta T$$ where $\Delta V$ is the change in volume, $V_0$ is the initial volume, $\beta$ is the coefficient of volume expansion for copper, and $\Delta T$ is the change in temperature. For solids, $\beta \approx 3\alpha$, where $\alpha$ is the linear expansion coefficient. 3. **Known values:** - Initial temperature $T_0 = 20.0^\circ C$ - Volume increase $\frac{\Delta V}{V_0} = 0.150\% = 0.00150$ - Coefficient of linear expansion for copper $\alpha = 16.5 \times 10^{-6} \, ^\circ C^{-1}$ 4. **Calculate $\beta$:** $$\beta = 3 \alpha = 3 \times 16.5 \times 10^{-6} = 49.5 \times 10^{-6} \, ^\circ C^{-1}$$ 5. **Find $\Delta T$:** $$0.00150 = 49.5 \times 10^{-6} \times \Delta T$$ $$\Delta T = \frac{0.00150}{49.5 \times 10^{-6}}$$ 6. **Simplify:** $$\Delta T = \frac{0.00150}{49.5 \times 10^{-6}} = \frac{0.00150}{0.0000495}$$ $$\Delta T = 30.3^\circ C$$ 7. **Find final temperature:** $$T = T_0 + \Delta T = 20.0 + 30.3 = 50.3^\circ C$$ **Answer:** The volume of the copper cylinder will be 0.150% larger at approximately $50.3^\circ C$ than at $20.0^\circ C$.