Compute the *exact* volume, area, or length as directed. Square roots do not need to be simplified, but arithmetic does.

- A cone has a volume of \(32 \pi\) and a height of \(4\). What is the radius of the cone?

\( \text{radius} = \sqrt{24} = 2\sqrt{6} \) - An ice cream cone is constructed from a cone and a hemisphere. The surface area of the hemisphere is \(50\pi\) and the cone height is \(7\). What is the volume of the ice cream cone and ice cream together?

\( \text{radius} = 5 \\ \text{volume} = \frac{2}{3}\pi 5^3 + \frac{1}{3} \cdot \pi 5^2 \cdot 7 = \frac{250 \pi}{3} + \frac{175\pi}{3} = \frac{425\pi}{3} = 141.{\overline 6} \pi \) - What is the volume of a triangular pyramid with base edges of length \(6\) and pyramid height of \(\sqrt{3}\)?

\( \frac{1}{3} \left( \frac{1}{2} 6 \sqrt{27} \right) \sqrt{3} = \sqrt{81} = 9 \) - What is the volume of a square pyramid with base edges of length \(6\) and
**pyramid height**of \(5\)?

\( \frac{1}{3} (6 \cdot 6) 5 = 60 \) - What is the volume of a square pyramid with base edges of length \(6\) and
**slant height**of \(5\)?

\( \text{pyramid height} = 4 \\ \text{volume} = \frac{1}{3} (6 \cdot 6) 4 = 48 \) - What is the volume of a square pyramid with base edges of length \(6\) and
**lateral edge length**of \(5\)?

\( \text{slant height} = 4 \\ \text{pyramid height} = \sqrt{7} \\ \text{volume} = \frac{1}{3} (6 \cdot 6) \sqrt{7} = 12 \sqrt{7} \) - What is the volume of a regular hexagonal prism with base sides of \(3\) and height \(7\)?

(Note: a regular hexagon can be split into six equilateral triangles)

\( (6 \cdot \frac{1}{2} 3 \sqrt{6.75}) 7 = 63 \sqrt{6.75} = 63 \sqrt{\frac{27}{4}} = \frac{189 \sqrt{3}}{2}\) - A silo is constructed from a cylinder with a hemisphere on top. What is the volume of a silo with a total height of \(10\) and base area of \(16\pi\)?

\( \text{radius} = 4 \\ \text{volume} = 96\pi + \frac{128}{3}\pi = \frac{416}{3}\pi \)

- A cylinder has a diagonal of \(10\) (that goes through the center of the cylinder) and a diameter of \(6\). What is the
**volume**and**surface area**of the cylinder?

\( \text{volume} = 72\pi \\ \text{surface area} = 66\pi \) - Consider a hemisphere inscribed in a cylinder. The cylinder has volume \(64\pi\). What is the volume of the hemisphere?

\( \text{radius} = 4 \\ \text{volume} = \frac{128}{3}\pi \) - A tube (in the shape of a cylinder) is made of thin material. The height is twice the diameter and the opening of the tube is half the length of the width. The tube is 4 feet long. What is the volume of the material used?

\( R = 1; r = \frac{1}{2}; \pi 1^2 4 - \pi \left(\frac{1}{2}\right)^2 4 = 3 \pi \) - What is the volume of an oblique (aka. skewed) regular hexagonal prism with base sides of \(3\) and height \(7\)?

same as previous, skewness doesn't change volume: \( 63 \sqrt{6.75} = 63 \sqrt{\frac{27}{4}} = \frac{189 \sqrt{3}}{2}\) - Mrs. Harris gave each child in her class a sheet of 8.5" x 11" paper and some sticky tape. She asked them to first roll the paper and make a cylinder. George and Eloise did as they were told, but they got two differently shaped cylinders. Mrs. Harris told them to determine with cylinder held the grater volume. George said, "mine holds the most, since it is taller." Eloise said, "No, mine holds more - its wider." Laura said, "They must both hold the same amount since they were made from the same size sheet of paper." Who is right? How can you tell?

\( V_{taller} = \pi (\frac{8.5}{2 \pi})^2 \cdot 11 = \frac{198.6875}{\pi} \\ V_{shorter} = \pi (\frac{11}{2 \pi})^2 \cdot 8.5 = \frac{257.125}{\pi} \)