Rachel,an engineering student,was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is $3\, cm$ and its total length is $12\, cm$. If each cone has a height of $2\, cm$,find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.) Unless stated otherwise,take $\pi = \frac{22}{7}$. (in $cm^3$)

$A$ fez,the cap used by the Turks,is shaped like the frustum of a cone (see $Fig.$). If its radius on the open side is $10 \, cm$,radius at the upper base is $4 \, cm$ and its slant height is $15 \, cm$,find the area of material used for making it.

Rasheed got a playing top $(lattu)$ as his birthday present,which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere (see figure). The entire top is $5 \, cm$ in height and the diameter of the top is $3.5 \, cm$. Find the area he has to colour (in $cm^2$). (Take $\pi = \frac{22}{7}$)

$A$ metallic sphere of radius $4.2 \, cm$ is melted and recast into the shape of a cylinder of radius $6 \, cm$. Find the height of the cylinder. (in $cm$) [Take $\pi = \frac{22}{7}$]

$A$ vessel is in the form of an inverted cone. Its height is $8 \, cm$ and the radius of its top,which is open,is $5 \, cm$. It is filled with water up to the brim. When lead shots,each of which is a sphere of radius $0.5 \, cm$,are dropped into the vessel,one-fourth of the water flows out. Find the number of lead shots dropped in the vessel. [Take $\pi = \frac{22}{7}$.]

Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half cylinder (see figure). If the base of the shed is of dimension $7\,m \times 15\,m,$ and the height of the cuboidal portion is $8\,m,$ find the volume of air that the shed can hold. Further,suppose the machinery in the shed occupies a total space of $300\,m^3,$ and there are $20$ workers,each of whom occupy about $0.08\,m^3$ space on an average. Then,how much air is in the shed? (in $m^3$) (Take $\pi = \frac{22}{7}$)