Mensuration：練習問題

A solid cylinder has a base radius of \(r\) and a height of \(h\). A hole in the shape of a right circular cone with the same radius \(r\) and the same height \(h\) is drilled into the cylinder from the top. If the volume of the remaining solid is \(128\pi\) cm\(^{3}\) and the height is twice the radius, find the value of \(r\).

Two solid spheres are made of the same material. The surface area of the first sphere is \(S\) and its mass is \(M\). If the surface area of the second sphere is \(4S\), find the mass of the second sphere in terms of \(M\).

A right circular cone has a base radius of \(r\) and a slant height of \(l\). A smaller cone is removed from the top of this cone by a plane parallel to the base. If the height of the removed cone is exactly half the height of the original cone, find the ratio of the volume of the remaining frustum to the volume of the original cone.

A cylinder has a radius of 3 cm and a height of 10 cm. Calculate the volume of the cylinder, giving your answer in cm\(^3\) in terms of \(\pi\).

The diagram shows a sector of a circle with radius \(r\) cm and center \(O\). The angle of the sector is \(60^{\circ}\). Given that the area of the sector is \(24\pi\) cm\(^2\), calculate the perimeter of the sector. Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are integers.

In the diagram, a solid metal cylinder has a radius of 5 cm and a height of 12 cm.

(a) Calculate the volume of the cylinder, giving your answer in terms of \(\pi\).
(b) The cylinder is melted down and recast into a solid sphere. Calculate the radius of the sphere, giving your answer to 3 significant figures.