Exercise 13.2

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $\mathrm{}1\mathrm{}\mathrm{c}\mathrm{m}$ and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\mathrm{\pi }$.

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 aluminum sheet. The diameter of the model is $3 \mathrm{c}\mathrm{m}$ and its length is $12 \mathrm{c}\mathrm{m}$. If each cone has a height of $2 \mathrm{c}\mathrm{m}$, 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.)

A $\mathrm{gulabjamun}$, contains sugar syrup up to about $30\mathrm{\%}$ of its volume. Find approximately how much syrup would be found in $45 \mathrm{gulabjamuns}$, each shaped like a cylinder with two hemispherical ends with length $\mathrm{}5\mathrm{}\mathrm{c}\mathrm{m}$ and diameter $\mathrm{}2.8\mathrm{}\mathrm{c}\mathrm{m}$ (see Fig.).

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 $15 \mathrm{c}\mathrm{m}$ by $10 \mathrm{c}\mathrm{m}$ by $3.5 \mathrm{c}\mathrm{m}$. The radius of each of the depressions is $0.5 \mathrm{c}\mathrm{m}$ and the depth is $1.4 \mathrm{c}\mathrm{m}$. Find the volume of wood in the entire stand (see figure).

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

A solid iron pole consists of a cylinder of height $220\mathrm{}\mathrm{c}\mathrm{m}$ and base diameter $24\mathrm{}\mathrm{c}\mathrm{m}$, which is surmounted by another cylinder of height $60\mathrm{}\mathrm{c}\mathrm{m}$ and radius $8\mathrm{}\mathrm{c}\mathrm{m}$. Find the mass of the pole, given that $\mathrm{}1\mathrm{}{\mathrm{c}\mathrm{m}}^3$ of iron has approximately $\mathrm{}8\mathrm{g}$ mass. (Use $\mathrm{\pi }=3.14$)

A solid consisting of a right circular cone of height $120 \mathrm{c}\mathrm{m}$ and radius $60 \mathrm{c}\mathrm{m}$ standing on a hemisphere of radius $60 \mathrm{c}\mathrm{m}$ is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60 \mathrm{c}\mathrm{m}$ and its height is $180 \mathrm{c}\mathrm{m}$.

A spherical glass vessel has a cylindrical neck $8 \mathrm{c}\mathrm{m}$ long, $2 \mathrm{c}\mathrm{m}$ in diameter; the diameter of the spherical part is $8.5 \mathrm{c}\mathrm{m}$. By measuring the amount of water it holds, a child finds its volume to be $345 \mathrm{c}{\mathrm{m}}^3$. Check whether she is correct, taking the above as the inside measurements, and $\mathrm{\pi }=3.14$.