Geometry

Volume - Problem Solving

If the above diagram consists of a hemi-sphere and a circular cone with $$r=9\text{ cm}$$ and $$h=9\text{ cm},$$ what is its volume?

The above diagram is not drawn to scale.

$$18$$ rigid balls of radius $$4\text{ cm}$$ are closely packed in a rectangular cuboid, as shown above. What is the ratio between the total volume occupied by the $$18$$ balls and the volume of the rectangular cuboid?

The cylinder on the left is filled with water, and the radius and height of the cylinder are $$r=3\text{ cm}$$ and $$h=10\text{ cm},$$ respectively. Now, we insert a rigid sphere with radius $$3\text{ cm}$$ into the cylinder causing the water to spill over. If all the water overflowing from the left cylinder can be contained in the right cylinder with radius $$3\text{ cm},$$ what is the minimum value of the height $$H$$ of the right cylinder in $$\text{cm} ?$$

The above diagram is a right pyramid with the following lengths: \begin{align} {\overline{AB}} &= {\overline{BC}} = {\overline{CD}} = {\overline{AD}} =4, \\ {\overline{OA}} &= {\overline{OB}} = {\overline{OC}} = {\overline{OD}} =2\sqrt{6}. \end{align} What is the volume of this pyramid?

Quadrilateral $$ABCD$$ is a square with side length $$18\text{ cm},$$ and $$E$$ and $$F$$ are the midpoints of $$\overline{BC}$$ and $$\overline{CD},$$ respectively. If we fold up the three corners $$B,$$ $$C$$ and $$D$$ to make a polyhedron, then what is the volume of the polyhedron in $$\text{cm}^3 ?$$

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