1. **State the problem:** We have a cylinder of height 15 cm with a hemisphere removed from the top such that the bottom of the hemisphere touches the center of the base of the cylinder. We need to find the volume of the remaining shape in terms of $\pi$. 2. **Identify the dimensions:** The height of the cylinder is 15 cm. Since the hemisphere's bottom touches the center of the base, the radius of the hemisphere equals the radius of the cylinder's base. Let this radius be $r$. 3. **Volume formulas:** - Volume of a cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of a sphere: $$V_{sphere} = \frac{4}{3} \pi r^3$$ - Volume of a hemisphere: $$V_{hemi} = \frac{1}{2} V_{sphere} = \frac{2}{3} \pi r^3$$ 4. **Determine the radius $r$:** The hemisphere's bottom touches the center of the base, so the height of the cylinder includes the radius of the hemisphere plus the remaining height above the hemisphere. Since the hemisphere is removed from the top, the height of the cylinder is the radius plus the remaining height. Given the total height is 15 cm, and the hemisphere radius is $r$, the height of the cylinder part remaining is $15 - r$. 5. **Calculate the volume of the cylinder:** $$V_{cyl} = \pi r^2 \times 15$$ 6. **Calculate the volume of the hemisphere removed:** $$V_{hemi} = \frac{2}{3} \pi r^3$$ 7. **Volume of the remaining shape:** $$V = V_{cyl} - V_{hemi} = \pi r^2 \times 15 - \frac{2}{3} \pi r^3$$ 8. **Simplify the expression:** $$V = \pi r^2 \times 15 - \frac{2}{3} \pi r^3 = \pi r^2 \left(15 - \frac{2}{3} r \right)$$ **Final answer:** $$\boxed{V = \pi r^2 \left(15 - \frac{2}{3} r \right)}$$ This is the volume of the shape in terms of $\pi$ and $r$. Since the problem does not provide the radius $r$, this is the most simplified form possible.