1. **Problem statement:** We want to find the work function $\phi$ and its uncertainty $\Delta \phi$ from the equation $$E = h \nu - \phi$$ given the measured values with uncertainties: $$\nu = (4.41 \pm 0.05) \times 10^{14} \text{ Hz}$$ $$E = (0.41 \pm 0.03) \times 10^{-19} \text{ J}$$ $$h = 6.63 \times 10^{-34} \text{ Js}$$ 2. **Rearranging the formula for $\phi$:** $$\phi = h \nu - E$$ 3. **Substitute the given values:** Calculate $h \nu$: $$h \nu = 6.63 \times 10^{-34} * 4.41 \times 10^{14} = 6.63 * 4.41 \times 10^{-34 + 14} = 29.2383 \times 10^{-20} = 2.92383 \times 10^{-19}$$ Now calculate $\phi$: $$\phi = 2.92383 \times 10^{-19} - 0.41 \times 10^{-19} = (2.92383 - 0.41) \times 10^{-19} = 2.51383 \times 10^{-19}$$ Rounded to 4 significant figures: $$\phi \approx 2.514 \times 10^{-19} \text{ J}$$ 4. **Partial derivatives for error propagation:** From $$\phi = h \nu - E$$, the partial derivatives are: $$\frac{\partial \phi}{\partial \nu} = h$$ $$\frac{\partial \phi}{\partial E} = -1$$ 5. **Calculate uncertainty $\Delta \phi$ using Gaussian error propagation:** $$\Delta \phi = \sqrt{\left(\frac{\partial \phi}{\partial \nu} \Delta \nu \right)^2 + \left(\frac{\partial \phi}{\partial E} \Delta E \right)^2} = \sqrt{(h \Delta \nu)^2 + (-1 * \Delta E)^2}$$ Calculate each term: $$h \Delta \nu = 6.63 \times 10^{-34} * 0.05 \times 10^{14} = 6.63 * 0.05 \times 10^{-34 + 14} = 0.3315 \times 10^{-20} = 3.315 \times 10^{-21}$$ $$\Delta E = 0.03 \times 10^{-19} = 3.0 \times 10^{-21}$$ Now: $$\Delta \phi = \sqrt{(3.315 \times 10^{-21})^2 + (3.0 \times 10^{-21})^2} = \sqrt{1.099 \times 10^{-41} + 9.0 \times 10^{-42}} = \sqrt{1.999 \times 10^{-41}} = 4.47 \times 10^{-21}$$ Rounded to 1 significant figure: $$\Delta \phi \approx 4 \times 10^{-21} \text{ J}$$ 6. **Final result:** $$\phi = (2.514 \pm 0.004) \times 10^{-19} \text{ J}$$