1. **State the problem:** We need to determine the space occupied by a water bottle, modeled mathematically. 2. **Identify the shape:** The bottle is roughly a vertical cylinder with rounded horizontal ribs and a narrower neck. For simplification, we model it as a cylinder plus a smaller cylinder (neck). 3. **Formula for volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 4. **Model the main body:** Let the main cylindrical body have radius $r_1$ and height $h_1$. 5. **Model the neck:** Let the neck have radius $r_2$ and height $h_2$. 6. **Total volume:** $$V_{total} = \pi r_1^2 h_1 + \pi r_2^2 h_2 = \pi (r_1^2 h_1 + r_2^2 h_2)$$ 7. **Interpretation:** This volume represents the space the bottle occupies in any context, assuming it is a solid object. 8. **Additional considerations:** The ribs add surface texture but negligible volume change; the base is flat and slightly wider, which can be approximated by adjusting $r_1$. 9. **Summary:** By measuring or estimating $r_1$, $h_1$, $r_2$, and $h_2$, we can calculate the bottle's volume and thus the space it occupies mathematically. **Final answer:** The space occupied by the water bottle is approximately $$V = \pi (r_1^2 h_1 + r_2^2 h_2)$$ where $r_1, h_1$ are the radius and height of the main body, and $r_2, h_2$ those of the neck.