Manipur Board Solutions for Chapter: Mensuration, Exercise 5: EXERCISE 12.5

 The radii of the bases of two solid metallic cones of same height $\mathrm{h}$ are ${\mathrm{x}}_{1 }\& {\mathrm{x}}_2$. If the two cones are melted together and recast into a cylinder of height $\mathrm{h}$, then show that the radius of the base of the cylinder is $\sqrt{\frac{1}{3}\left({{\mathrm{x}}^2}_{1 }+{{ \mathrm{x}}^2}_2\right)}$.

    A solid metallic cylinder and another solid metallic cone have the same height $\mathrm{h}$ and the same radius $\mathrm{r}$. If the two solids are melted together and recast into a cylinder of radius $\frac{1}{2}\mathrm{r}$, prove that the height of the new cylinder is $\frac{16}{3} \mathrm{h}$.

Two solid metallic cuboids of dimensions $15 \mathrm{cm}\times 8 \mathrm{cm}\times 5 \mathrm{cm}$ and $20 \mathrm{cm}\times 5 \mathrm{cm}\times 4 \mathrm{cm}$, are melted together and recast into solid cubes each of side $5 \mathrm{cm}$. Find the number of solid cubes so formed.

 A sphere of diameter $6 \mathrm{cm}$ is dropped into a cylindrical vessel, partly filled with water. The diameter of the vessel is $12 \mathrm{cm}$. If the sphere is completely submerged, how high will the level of the water be raised?

A solid metallic cylinder of radius $14 \mathrm{cm}$ and height is $21 \mathrm{cm}$ is melted and recast into a number of spheres, each of radius $3.5 \mathrm{cm}$. Find the number of spheres so formed.

A spherical shell of lead, whose external diameter is $16 \mathrm{cm}$, is melted and recast into a cylinder of height $9 \frac{1}{3} \mathrm{cm}$ and diameter $16 \mathrm{cm}$. Find the internal diameter of the shell.

A vessel is in the form of an inverted cone of height $8 \mathrm{cm}$ and radius $6 \mathrm{cm}$. It is filled with water upto the rim. When lead shots, each of which is a sphere of radius $0.2 \mathrm{cm}$ are dropped into the vessel, one -sixth of the water flows out. Find the number of lead
shots dropped into the vessel.
 

 Water is flowing at the rate of $5 \mathrm{km}$ per hour through a pipe of diameter $14 \mathrm{cm}$ into a rectangular tank of base $30 \mathrm{m}\times 22 \mathrm{m}$. Find the time during which the level of water in the tank rises by $35 \mathrm{cm}$.(in $\mathrm{hours}$)