1) Find the total area of a sphere that will fit exactly inside a cube of volume \(4096\) cm\(^3\). (The sphere touches all \(6\) sides of the cube.)
2) Find the simplified ratio of the volumes of two cubes whose surface areas are \(18\) and \(50\).
3) Given a regular tetrahedron with volume \(1\) cm\(^3\) and a cube with volume \(1\) cm\(^3\), which object has smaller surface area? Explain your reasoning.
4) The diagonals of the faces of a rectangular prism are \(5,\), \(7\) & \(8\). Find the length of the diagonal of the prism.
5) The volume of a regular square pyramid is \(\Large\frac{64}{3}\) with a base edge of \(4\). What is the length of a lateral edge of the pyramid?
6) When a conical filter is empty, it lies flat as a quarter circle that is two layers thick. If the edge length is \(12.5\) cm, what is the height of the filter when it is filled with coffee grounds?
2) Find the simplified ratio of the volumes of two cubes whose surface areas are \(18\) and \(50\).
3) Given a regular tetrahedron with volume \(1\) cm\(^3\) and a cube with volume \(1\) cm\(^3\), which object has smaller surface area? Explain your reasoning.
4) The diagonals of the faces of a rectangular prism are \(5,\), \(7\) & \(8\). Find the length of the diagonal of the prism.
5) The volume of a regular square pyramid is \(\Large\frac{64}{3}\) with a base edge of \(4\). What is the length of a lateral edge of the pyramid?
6) When a conical filter is empty, it lies flat as a quarter circle that is two layers thick. If the edge length is \(12.5\) cm, what is the height of the filter when it is filled with coffee grounds?
7) A sugar cone has a height of \(11.7\) cm and a diameter of \(4.8\) cm. If the sugar cone is topped off with a spherical scoop of ice cream that all melts into the cone, how far up the cone will the melted ice cream go?
8) Suppose the frustum has a volume of \(28\pi\) cm\(^2\). Find the height (\(h\)) of the frustum.