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 $\pi = \frac{22}{7}$)

$A$ 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.

Metallic spheres of radii $6 \, cm$,$8 \, cm$,and $10 \, cm$,respectively,are melted to form a single solid sphere. Find the radius of the resulting sphere (in $cm$).

$A$ pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are $15 \, cm$ by $10 \, cm$ by $3.5 \, cm$. The radius of each of the depressions is $0.5 \, cm$ and the depth is $1.4 \, cm$. Find the volume of wood in the entire stand (in $cm^3$). [Take $\pi = \frac{22}{7}$]

$A$ container,opened from the top and made up of a metal sheet,is in the form of a frustum of a cone of height $16 \, cm$ with radii of its lower and upper ends as $8 \, cm$ and $20 \, cm$,respectively. Find the cost of the milk which can completely fill the container,at the rate of $Rs. \, 20$ per litre. Also,find the cost of the metal sheet used to make the container,if it costs $Rs. \, 8$ per $100 \, cm^2$. (Take $\pi = 3.14$)

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 $cm^3$)