1. **State the problem:** We need to find the approximate increase in the volume of a cylinder when its radius increases from 4 cm to 4.20 cm, given the height is 10 cm. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Calculate the initial volume:** $$V_1 = \pi (4)^2 (10) = 160\pi$$ 4. **Calculate the new volume with increased radius:** $$V_2 = \pi (4.20)^2 (10) = \pi (17.64)(10) = 176.4\pi$$ 5. **Find the increase in volume:** $$\Delta V = V_2 - V_1 = 176.4\pi - 160\pi = 16.4\pi$$ 6. **Approximate the increase using $\pi \approx 3.1416$:** $$\Delta V \approx 16.4 \times 3.1416 = 51.5$$ **Final answer:** The approximate increase in volume is $51.5$ cubic centimeters.