Karnataka Board Solutions for Chapter: Solved Examples, Exercise 13: Exercise 13

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

A juice seller was serving his customers using glasses as shown in the figure. The inner diameter of the cylindrical glass was $5 \mathrm{cm}$, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was $10 \mathrm{cm}$, find the apparent capacity of the glass and its actual capacity. (Use $\mathrm{\pi }=3.14$.)

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is $2 \mathrm{cm}$ and the diameter of the base is $4 \mathrm{cm}$. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. (Take $\mathrm{\pi }=3.14$)

Selvi's house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (an underground tank) which is in the shape of a cuboid. The sump has dimensions $1.57 \mathrm{m}\times 1.44 \mathrm{m}\times 95 \mathrm{cm}$. The overhead tank has its radius $60 \mathrm{cm}$ and height $95 \mathrm{cm}$. Find the height of the water left in the sump after the overhead tank has been completely filled with water from the sump which had been full. Compare the capacity of the tank with that of the sump. (Use $\mathrm{\pi }=3.14$)

A copper rod of diameter $1 \mathrm{cm}$ and length $8 \mathrm{cm}$ is drawn into a wire of length $18 \mathrm{m}$ of uniform thickness. Find the thickness of the wire.

The radii of the ends of a frustum of a cone $45 \mathrm{cm}$ high are $28 \mathrm{cm}$ and $7 \mathrm{cm}$. Find its volume, the curved surface area and the total surface area (Take $\mathrm{\pi }=\frac{22}{7}$).

Hanumappa and his wife Gangamma are busy making jaggery out of sugarcane juice. They processed the sugarcane juice to make the molasses, which is poured into moulds in the shape of a frustum of a cone having the diameters of its two circular faces as $30 \mathrm{cm}$ and $35 \mathrm{cm}$ and the vertical height of the mould is $14 \mathrm{cm}$. If each ${\mathrm{cm}}^3$ of molasses has mass about $1.2 \mathrm{g}$, find the mass of the molasses that can be poured into each mould.$\left(\mathrm{Take} \mathrm{\pi }=\frac{22}{7}\right)$

An open metal bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The diameters of the two circular ends of the bucket are $45 \mathrm{cm}$ and $25 \mathrm{cm}$, the total vertical height of the bucket is $40 \mathrm{cm}$ and that of the cylindrical base is $6 \mathrm{cm}$. Find the area of the metallic sheet used to make the bucket, where we do not take into account the handle of the bucket. Also, find the volume of water the bucket can hold Take $\mathrm{\pi }=\frac{22}{7}$.