Bidyut Guha Thakurta and Amitava Mitra Solutions for Exercise 1: EXERCISE 5

The heights of a sphere and a right circular cylinder are the same. If their volumes are equal, the ratio of square of the radii of the sphere and the cylinder is _____.

By melting a solid right circular cylinder, a solid right circular cone of same height as the cylinder is made. The ratio of the radii of the cylinder and the cone is  _____$: \sqrt{3}$

The length of a cylinder is $\mathrm{l}$ and the radius of its cross-section is $\mathrm{r}$. If the volume be equal to that of the sphere of the same radius, then the relation between $\mathrm{r}$ and $\mathrm{l}$ is _____.

The number of spherical solids each $2 \mathrm{cm}$ in radius that can be cast from a solid iron cylinder of radius $4 \mathrm{cm}$ and height $10 \mathrm{cm}$, is $15$.

The two radii and the volumes of a sphere and a right circular cylinder are the same. The ratio of the height and the radius of the cylinder is $\frac{3}{4}$.

A rectangular paper sheet is of length $\mathrm{l}$. It is folded to form a hollow right circular cylinder such that the breadth of the paper becomes the height of the cylinder. The radius of the greatest cylinder is $\frac{\mathrm{l}}{2\mathrm{\pi }}$.

The percentage increase in the volume of a solid hemisphere if its radius is doubled, is $300\%$.

A sphere of the greatest size is kept inside a cubical box. If the internal dimension of the box be x unit, then the volume of sphere is $\frac{{\mathrm{\pi x}}^3}{6}$ cu. units.