Three-dimensional puzzles are popular toys for adolescents, especially since the Rubik's cube. The wooden puzzle shown here has only $10$ pieces but is very difficult to solve. The puzzle comes with the pieces arranged in the shape of a cube, and the aim is to rearrange them to make a square pyramid. If the puzzle contains $9 {\text{cm}}^3$ of wood, find the dimensions of the pyramid. 

New Zealand, which is known for its adventurous activities, is the home of "zorbing". This involves rolling down a hill in an inflatable ball. The zorb is actually a sphere within a sphere, with a layer of air in between to help absorb shocks. Zorbs can hold up to three people and zorbers can even choose the "aqua option", where the inner sphere is partly filled with water. This makes for a refreshing ride down the hill on a hot day. If the outer sphere has a  diameter of $3$ m and the inner sphere has a  diameter of $2$ m, If the volume of the air layer between the inner and outer spheres (The plastic material for each sphere is approximately $0.8$ mm thick, which can be ignored in these calculations.) rounded to $3$ s.f. is $V {\text{m}}^3$ then find $V$. $(\text{Take} \pi =22/7)$

Find the volume and surface area of the following $3\text{D}$ shape. Round your answer to the nearest tenth. $\left(\text{Take} \pi =22/7\right)$

Find the volume and surface area of the following $3\text{D}$ shape. Round your answer to the nearest tenth. $\left(\text{Take} \pi =22/7\right)$

Find the volume and surface area of the following $3\text{D}$ shape. Round your answer to the nearest tenth. $\left(\text{Take} \pi =22/7\right)$

Specialized paint has been designed to protect storage silos against corrosion and to prolong their life. First the silo is painted with a layer of rust-resistant primers and then with two layers of the specialized paint. The height of the cylindrical part of one agricultural silo is $17$ m and the total height of the silo is $20$ m. The diameter of the silo is $15$ m. If the surface area to protect the five silos is $S {\text{m}}^2$ then find $S$. 

The Morning Glory cloud formation occurs very predictably on the north coast of Australia. Although its causes are not known precisely, it forms mostly in October over the Gulf of Carpentaria during a process when sea breezes from the west meet breezes from the east. To study such phenomena, scientists use a tool called a radiosonde, attached to a weather balloon (which is spherical!). These cylindrical cloud formations can be $1000$ km long with a radius of $150$ m. Moving at speeds up to $60$ km/h, there is a lot of cloud moving overhead. If the surface area of the Morning Glory cloud formation, rounded to the nearest hundredth is $S {\mathrm{km}}^2$ then find $S$. (Take $=22/7$)

The Morning Glory cloud formation occurs very predictably on the north coast of Australia. Although its causes are not known precisely, it forms mostly in October over the Gulf of Carpentaria during a process when sea breezes from the west meet breezes from the east. To study such phenomena, scientists use a tool called a radiosonde, attached to a weather balloon (which is spherical!). These cylindrical cloud formations can be $1000$ km long with a radius of $150$ m. Moving at speeds up to $60$ km/h, there is a lot of cloud moving overhead. If the cloud formation were only half as long, by what factor would that change the surface area? Justify your answer with calculations. 

A common product found in many restaurants and kitchens is the pepper grinder. It allows users to grind their own pepper, keeping it as fresh as possible until used. Pepper grinders can come in a wide range of shapes and sizes. One product is in the shape of a cylinder with a height of $14$ cm and a radius of $5$ cm. Another is in shape of a ball (sphere). If the radius of a ball pepper grinder if it has to contain the same volume of pepper as the cylinder rounded to nearest tenths is $R$ cm then find $R$.