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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