CBSE Class 10 Mathematics Surface Areas And Volume VBQs Set H

I. Very Short Answer Type Questions 

Multiple Choice Questions (MCQs) Choose the correct answer from the given options:

Question. A cylindrical pencil sharpened at one edge is the combination of
(a) a cone and a cylinder
(b) frustum of a cone and a cylinder
(c) a hemisphere and a cylinder
(d) two cylinders
Answer: (a) a cone and a cylinder

Question. A surahi is the combination of
(a) a sphere and a cylinder
(b) a hemisphere and a cylinder
(c) two hemispheres
(d) a cylinder and a cone
Answer: (a) a sphere and a cylinder

Question. In a right circular cone, the cross-section made by a plane parallel to the base is a
(a) Triangle
(b) Circle
(c) Square
(d) None of the options
Answer: (b) Circle

Question. If two solid hemispheres of same base radius \( r \) are joined together along their bases, then curved surface area of this new solid is
(a) \( 4\pi r^2 \)
(b) \( 3\pi r^2 \)
(c) \( 2\pi r^2 \)
(d) \( \pi r^2 \)
Answer: (a) \( 4\pi r^2 \)

Question. Assertion-Reason Type Questions
In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion (A): The number of coins 1.75 cm in diameter and 2 mm thick is formed from a melted cuboid 10 cm × 5.5 cm × 3.5 cm is 400.
Reason (R): Volume of a cylinder = \( \pi r^2 h \) cubic units and volume of cuboid = \( (l \times b \times h) \) cubic units.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): Number of spherical balls that can be made out of a solid cube of lead whose edge is 44 cm, each ball being 4 cm in diameter is 2541.
Reason (R): \( \text{Number of balls} = \frac{\text{Volume of lead}}{\text{Volume of one ball}} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Answer the following.

Question. What is the capacity of cylindrical vessel with the hemispherical bottom portion raised upwards?
Answer: \( \frac{\pi r^2}{3}[3h - 2r] \)

Question. A solid is hemispherical at the bottom and conical (of same radius) above it. If the surface areas of two parts are equal, then what is the ratio of its radius and the slant height of the conical part?
Answer: 1 : 2

II. Short Answer Type Questions-I   

Question. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Answer: Here, \( r = 3.5 \text{ cm} \). Height of conical part \( h = (15.5 - 3.5) \text{ cm} = 12.0 \text{ cm} \).
Slant height \( l = \sqrt{(12)^2 + (3.5)^2} = \sqrt{144 + 12.25} = \sqrt{156.25} = 12.5 \text{ cm} \).
TSA of toy = CSA of cone + CSA of hemisphere \( = \pi rl + 2\pi r^2 = \pi r(l + 2r) \)
\( = \frac{22}{7} \times \frac{35}{10} (12.5 + 2 \times 3.5) \text{ cm}^2 = 11 \times (12.5 + 7) \text{ cm}^2 = 11 \times 19.5 \text{ cm}^2 = 214.5 \text{ cm}^2 \).

Question. A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the solid is 58 cm and the diameter of the cylinder is 28 cm. Find the total surface area of the solid. [Use \( \pi = \frac{22}{7} \)]
Answer: 6160 \( \text{cm}^2 \)

III. Short Answer Type Questions-II  

Question. A toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter of the base of the conical portion is 6 cm and its height is 4 cm. Determine the surface area of the toy. [Take \( \pi = 3.14 \)]
Answer: 103.62 \( \text{cm}^2 \)

Question. From a solid cylinder of height 12 cm and diameter of the base 10 cm a conical cavity of the same height and same diameter is hollowed out. Find the surface area of the remaining solid.
Answer: 660 \( \text{cm}^2 \)

Question. A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The height and the radius of the cylindrical part are 13 cm and 5 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if height of the conical part is 12 cm.
Answer: 770 \( \text{cm}^2 \)

Question. A gulab jamun when completely ready for eating contains sugar syrup upto about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns shaped like a cylinder with two hemispherical ends, if the complete length of each of the gulab jamuns is 5 cm and its diameter is 2.8 cm.
Answer: 337.88 \( \text{cm}^3 \)

Question. 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 the glass was 10 cm, find its actual capacity and its apparent capacity. (Use \( \pi = 3.14 \))
Answer: Apparent capacity = \( 196.25 \text{ cm}^3 \); Actual capacity = \( (196.25 - 32.71) = 163.54 \text{ cm}^3 \).

IV. Long Answer Type Questions 

Question. A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of cylinder. The diameter and height of cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, then find the total surface area and volume of rocket. (Use \( \pi = 3.14 \))
Answer: TSA = \( 301.44 \text{ cm}^2 \); Volume = \( 376.8 \text{ cm}^3 \).

Question. A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. Find the volume of water left in cylinder, if the radius of the cylinder is equal to the radius to the cone.
Answer: \( 1.584 \text{ m}^3 \)

Question. A solid is in the shape of a cone surmounted on a hemisphere. The radius of each of them being 3.5 cm and the total height of the solid is 9.5 cm. Find the volume of the solid.
Answer: \( 166.83 \text{ cm}^3 \)

Case Study Based Questions

I. The Great Stupa at Sanchi is one of the oldest stone structures in India, and an important monument of Indian Architecture. It was originally commissioned by the emperor Ashoka in the 3rd century BCE. Its nucleus is a simple hemispherical brick structure built over the relics of the Buddha. It is a perfect example of combination of solid figures. A big hemispherical dome with a cuboidal structure mounted on it. (Take \( \pi = \frac{22}{7} \))

Question. The volume of the hemispherical dome if the height of the dome is 21 m, is
(a) 19404 cu. m
(b) 2000 cu. m
(c) 15000 cu. m
(d) 19000 cu. m
Answer: (a) 19404 cu. m

Question. The formula to find the volume of sphere is
(a) \( \frac{2}{3}\pi r^3 \)
(b) \( \frac{4}{3}\pi r^3 \)
(c) \( 4\pi r^2 \)
(d) \( 2\pi r^2 \)
Answer: (b) \( \frac{4}{3}\pi r^3 \)

Question. The cloth required to cover the hemispherical dome if the radius of its base is 14 m is
(a) 1222 sq. m
(b) 1232 sq. m
(c) 1200 sq. m
(d) 1400 sq. m
Answer: (b) 1232 sq. m

Question. The total surface area of the combined figure, i.e. hemispherical dome with radius 14 m and cuboidal shaped top with dimensions 8 m × 6 m × 4 m is
(a) 1200 sq. m
(b) 1232 sq. m
(c) 1392 sq. m
(d) 1932 sq. m
Answer: (c) 1392 sq. m

Question. The volume of the cuboidal shaped top with dimensions mentioned in question 4, is
(a) \( 182.45 \text{ m}^3 \)
(b) \( 282.45 \text{ m}^3 \)
(c) \( 292 \text{ m}^3 \)
(d) \( 192 \text{ m}^3 \)
Answer: (d) \( 192 \text{ m}^3 \)