1. **State the problem:** We need to find the gage pressure inside an air bubble of radius 1.0 mm (0.001 m) submerged 10 cm (0.1 m) below the free surface of water at 20°C. 2. **Understand gage pressure:** Gage pressure is the pressure above atmospheric pressure. It is given by the hydrostatic pressure at that depth plus any additional pressure from surface tension if relevant. 3. **Calculate hydrostatic pressure:** Hydrostatic pressure at depth $h$ in a liquid of density $\rho$ under gravity $g$ is $$p_h = \rho g h.$$ For water at 20°C, the density $\rho \approx 998 \text{ kg/m}^3$, gravity $g = 9.81 \text{ m/s}^2$, and depth $h = 0.1 \text{ m}$. Thus, $$p_h = 998 \times 9.81 \times 0.1 = 978.438 \text{ Pa}.$$ 4. **Include pressure due to surface tension:** The inside pressure in a bubble is higher than the outside pressure by the amount $$\Delta p = \frac{2 \gamma}{r}$$ where $\gamma$ is the surface tension of water and $r$ is bubble radius. At 20°C, surface tension of water $\gamma \approx 0.0728 \text{ N/m}$. Given $r = 0.001 \text{ m}$, we have $$\Delta p = \frac{2 \times 0.0728}{0.001} = 145.6 \text{ Pa}.$$ 5. **Calculate total gage pressure inside the bubble:** This is hydrostatic pressure plus pressure from surface tension: $$p_{gage} = p_h + \Delta p = 978.438 + 145.6 = 1124.038 \text{ Pa}.$$ 6. **Final answer:** The gage pressure inside the air bubble is approximately $$\boxed{1124 \text{ Pa}}.$$