Class 10 - Chapter 12: Surface Areas and Volumes - Test 3

Section A (1 Mark each)

Q1. If a solid sphere of radius $r$ is melted and cast into the shape of a solid cone of base radius $R$ and height $r$, then the value of $R$ is:

Q2. Three solid metallic cubes of edges 6 cm, 8 cm, and 10 cm are melted and recast into a single solid cube. The edge of the new cube is:

Q3. A gilli in the game of gilli-danda is a combination of:

Q4. Assertion (A): The largest sphere that can be carved out of a cube of edge $a$ cm has radius $\frac{a}{2}$ cm.
Reason (R): The diameter of the largest sphere inscribed in a cube is equal to the edge of the cube.

Section B (2 Marks each)

Q5. 500 persons are taking a dip into a cuboidal pond which is 80 m long and 50 m broad. What is the rise of water level in the pond, if the average displacement of the water by a person is $0.04\text{ m}^3$?

Q6. A metallic sphere of radius 10.5 cm is melted and recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

Q7. A wall 270 cm long, 300 cm high and 350 cm thick is made of bricks, each measuring $22.5\text{ cm} \times 11.25\text{ cm} \times 8.75\text{ cm}$. If $\frac{1}{8}$ of the total volume of the wall consists of mortar, find the number of bricks required to make the wall.

Section C (3 Marks each)

Q8. Water flows at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank in half an hour is 3.15 m, find the internal diameter of the pipe.

Q9. A wooden toy is in the shape of a cone mounted on a hemisphere with the same base radius. The radius of the base of the cone is 1.75 cm and total height of the toy is 5 cm. Find the total surface area of the toy to be painted. (Take $\pi = \frac{22}{7}$)

Section D (5 Marks)

Q10. A solid right circular cone of radius 5 cm and height 8 cm is full of water. A number of lead shots, each of spherical shape of diameter 1 cm, are dropped into the cone such that one-fourth of the water overflows. Find the number of lead shots dropped into the cone.

Section E (Case Study - 4 Marks)

A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 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 cm (Take $\pi = 3.14$):

(i) Find the apparent capacity of the glass.

(ii) Find the actual capacity of the glass.

(iii) Find the volume of the raised hemispherical portion.