Two transmitters are located at $A(2,-1,4)$ and $B(-4,3,8)$. A relay is to be placed at a point $R(0,0,z)$ on the $z$-axis. All coordinates are measured in kilometres.

A sector of a circle has radius $15$ cm and central angle $216^\circ$. Its two straight edges are joined to form the curved surface of a right cone.

A hollow spherical ornament has outer radius $10$ cm and uniform wall thickness $1.2$ cm.

A sealed capsule consists of a right cylinder of radius $2$ cm with a hemisphere of the same radius attached at each end. The total internal volume is $100\pi\text{ cm}^3$.

An open cylindrical container has radius $r$ cm, height $h$ cm and volume $500\text{ cm}^3$. Its surface area, excluding the open top, is $A\text{ cm}^2$.

A right pyramid has a rectangular base measuring $10$ m by $6$ m. Its apex is $8$ m vertically above the centre of the base.

A cable joins a point $A(2,-3,0)$ on a horizontal platform to a point $B(8,5,12)$ above it. The platform is the plane $z=0$, and coordinates are measured in metres. The orthogonal projection of $B$ onto the platform is $C$.

A grain silo consists of a right circular cylinder of radius $4$ m and height $9$ m, with a right conical roof of the same radius. The slant height of the roof is $5$ m. The circular floor is not painted.

A drone travels in a straight line from $A(-3,2,1)$ to $B(5,8,7)$. Coordinates are measured in kilometres, and the plane $z=0$ represents horizontal ground.

A frozen dessert consists of a hemisphere of radius $3$ cm joined to the circular opening of a right cone of the same radius. The cone has perpendicular height $10$ cm.

A temporary exhibition tent is a right square-based pyramid. The square floor has side length $6$ m and the apex is $4$ m vertically above its centre. The floor is not covered by fabric.

An observatory consists of a vertical right cylinder of radius $7$ m and height $4$ m, with a hemispherical dome of the same radius. The circular floor is included when calculating internal volume but is not painted.

A marine marker is modelled as a right cone joined to a solid hemisphere along a common circular face. The common radius is $2$ m and the cone has perpendicular height $3$ m. The joined circular face is internal.

A planter is modelled as a frustum of a right cone. Its open top has radius $6$ m, its circular base has radius $3$ m, and its perpendicular height is $8$ m. When extended, the sloping sides meet at the vertex of a cone.

A storage shelter is a right prism of length $10$ m. Its constant cross-section is an isosceles trapezium with parallel sides $3$ m and $5$ m and perpendicular height $2.4$ m.

A cylindrical candle has radius $3$ cm and height $12$ cm. A conical cavity of radius $2$ cm and perpendicular depth $5$ cm is removed from the centre of its top.

A solid metal sphere of radius $12$ cm is melted and recast into $n$ identical smaller spheres of radius $r$ cm. No metal is lost.

A cubic display case has side length $8$ m. Relative to one lower corner as the origin, its vertices have coordinates from $0$ to $8$. Point $P(4,0,8)$ is the midpoint of an upper edge and $Q(8,8,0)$ is a lower vertex.

An exhibition building consists of a right circular cylinder of radius $r$ metres and height $h$ metres, with a hemispherical roof of the same radius. The total height of the building is $18$ m. For practical reasons, the cylindrical section must have height at least $6$ m.

A drone travels in a straight line from $A(1,2,0)$ to $B(9,8,12)$, where coordinates are measured in kilometres and the plane $z=0$ represents level ground. The orthogonal projection of $B$ onto the ground is $C$.

A glass skylight is modelled as a right square-based pyramid. Its base has side length $12$ m and the perpendicular slant height of each triangular face is $10$ m. The apex lies vertically above the centre of the base.

A plant pot is modelled as a frustum formed by removing a smaller cone from a right cone. The original cone has base radius $6$ cm and height $15$ cm. The removed cone has height $5$ cm. The top of the pot is open.

A right pyramid has a rectangular base measuring $14$ m by $10$ m. Its apex is $9$ m vertically above the centre of the base.

An aircraft hangar is a right prism of length $30$ m. Its cross-section consists of a rectangle of width $12$ m and height $4$ m, topped by a semicircle of radius $6$ m.

A star is modelled as a sphere of radius $R$. Its total power output is assumed to spread uniformly over spherical surfaces. At distance $d$ from the centre, the intensity is modelled by $I=\frac{L}{4\pi d^2}$, where $L$ is constant.

A closed cylindrical container has radius $r$ cm, height $h$ cm and fixed volume $750\text{ cm}^3$. Its total surface area is $A\text{ cm}^2$.

A right square-based pyramid has base side length $x$ cm and perpendicular height $(x+2)$ cm. Its volume is $200\text{ cm}^3$.

A decorative block is a frustum of a right square-based pyramid. Its lower square has side length $12$ cm, its upper square has side length $x$ cm, and its perpendicular height is $9$ cm. Its volume is $684\text{ cm}^3$.

A sealed underwater chamber consists of a right cylinder with a hemisphere attached at each end. Its total length is $20$ m and its internal volume is $1000\text{ m}^3$. The common radius is $r$ m. Assume the chamber walls have negligible thickness, so the internal and external dimensions are the same.

A grain silo consists of a right circular cylinder of radius $r$ m and height $h$ m, topped by a right cone of the same radius and perpendicular height $r$ m. The total internal volume is $500\pi\text{ m}^3$. The base is not included in the area to be coated.

A spherical display of radius $5$ m is placed at the centre of a cubical hall of side length $10$ m. Eight identical smaller spheres are placed in the corners. Each smaller sphere is tangent to the three faces meeting at its corner and tangent to the central sphere.

A pavilion roof is a frustum of a right square-based pyramid. The complete pyramid has base side length $10$ m and vertical height $12$ m. The roof is cut by a plane parallel to the base so that the top square has side length $4$ m.

An open hemispherical bowl has outer radius $6$ cm and uniform thickness $0.5$ cm. The material is modelled as the region between two concentric hemispheres. The flat annular rim is exposed.

A straight structural beam joins $A(-6,2,3)$ to $B(4,10,15)$ in a three-dimensional coordinate system measured in metres. A vertical partition is represented by the plane $x=0$.

A circular sheet of radius $20$ cm has a sector removed. The two remaining radial edges are joined to form the curved surface of a right cone. The slant height of the cone is therefore $20$ cm. A particular cone is required to have perpendicular height $12$ cm.

A right cone is placed inside a sphere of radius $10$ cm so that the cone's vertex and the circumference of its base lie on the sphere. The cone has perpendicular height $h$ cm and base radius $r$ cm.

A right circular cylinder is placed inside a sphere of radius $10$ cm. The cylinder is centred at the centre of the sphere. Its radius is $r$ cm and its total height is $h$ cm.

A family of right square-based pyramids has sloping edge length $15$ m. Each apex lies vertically above the centre of its square base. Let the base side length be $s$ m and the vertical height be $h$ m.

A rectangular observation chamber has dimensions $x$ m, $2x$ m and $z$ m. Its volume is $480\text{ m}^3$. A laser beam joins one lower vertex to the opposite upper vertex and makes an angle of $45^\circ$ with the base.