CBSE Class 10 Mathematics Surface Areas And Volume VBQs Set G

Multiple Choice Questions (MCQs)

Question. The radius of a sphere (in cm) whose volume is \( 12\pi \text{ cm}^3 \), is
(a) 3
(b) \( \sqrt{3} \)
(c) \( 3^{2/3} \)
(d) \( 3^{1/3} \)
Answer: (c) \( 3^{2/3} \)

Question. If the surface area of a sphere is \( 144\pi \), then its radius is
(a) 6 cm
(b) 8 cm
(c) 12 cm
(d) 10 cm
Answer: (a) 6 cm

Question. The edge of a cube whose volume is equal to that of a cuboid of dimensions 8 cm × 4 cm × 2 cm is
(a) 6 cm
(b) 4 cm
(c) 2 cm
(d) 8 cm
Answer: (b) 4 cm

Question. If the radii of two spheres are in the ratio 2 : 3, then the ratio of their respective volumes is
(a) 8 : 27
(b) 3 : 5
(c) 7 : 24
(d) 5 : 14
Answer: (a) 8 : 27

Question. The edge of a cube whose volume is \( 8x^3 \) is
(a) \( x \)
(b) \( 2x \)
(c) \( 4x \)
(d) \( 8x \)
Answer: (b) \( 2x \)

Question. If the volume of a 7 cm high right circular cylinder is \( 448\pi \text{ cm}^3 \), then its radius is equal to
(a) 2 cm
(b) 4 cm
(c) 6 cm
(d) 8 cm
Answer: (d) 8 cm

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): Total surface area of the cylinder having radius of the base 14 cm and height 30 cm is 3872 cm².
Reason (R): If \( r \) be the radius and \( h \) be the height of the cylinder, then total surface area \( = (2\pi rh + 2\pi r^2) \).
(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): If the height of a cone is 24 cm and diameter of the base is 14 cm, then the slant height of the cone is 15 cm.
Reason (R): If \( r \) is the radius and \( h \) the height of the cone, then slant height \( = \sqrt{h^2 + r^2} \).
(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: (d) Assertion (A) is false but reason (R) is true.

Answer the following.

Question. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
Answer: 9 units

Question. Total surface area of a cube is 216 cm². Find its volume.
Answer: 216 cm³

Question. If a solid right-circular cone of height 24 cm and base radius 6 cm is melted and recast in the shape of a sphere, find the radius of the sphere.
Answer: 6 cm

Question. Find the curved surface area of a right-circular cone of height 15 cm and base diameter 16 cm.
Answer: \( 136\pi \text{ cm}^2 \)

Question. Find the radius of the sphere whose surface area is \( 36\pi \text{ cm}^2 \).
Answer: 3 cm

II. Short Answer Type Questions-I 

Question. A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.
Answer: 2.744 cm

Question. Three solid metal cubes of edges 6 cm, 8 cm and 10 cm are melted and recasted into a single solid cube. Find the length of the edge of the cube so obtained.
Answer: 12 cm

Question. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.
Answer: 0.67 mm (approx)

Question. Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.
Answer: 12 cm

Question. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.
Answer: 2.5 m

III. Short Answer Type Questions-II 

Question. A solid sphere is melted and recasted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm, its height 24 cm and thickness 2 cm; find the radius of the sphere.
Answer: 6 cm

Question. A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?
Answer: Volume = 528 m³, Canvas required = 471.4 m²

Question. A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere
Answer: 6 cm

Question. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?
Answer: 400

Question. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm standing water is needed?
Answer: 562500 m²

Question. In a hospital, used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park?
Answer: 2.5 cm

Question. In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, find the rainfall in cm.
Answer: 2.5 cm

Question. Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively are melted to form a solid sphere. Find the radius of the resulting sphere.
Answer: 12 cm

IV. Long Answer Type Questions

Question. A well, whose diameter is 3 m, has been dug 21 m deep and the earth dug out is used to form an embankment 4 m wide around it. Find the height of the embankment.
Answer: 1.125 m (Note: OCR shows 1.125m based on previous example logic, checking text value: 1.125 m)

Question. A farmer connects a pipe of internal diameter 20 cm from the canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?
Answer: 100 minutes

Question. Selvi’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from an underground tank which is in the shape of a cuboid. The underground tank has dimensions 1.57 m × 1.44 m × 95 cm. The overhead tank has its radius of 60 cm and its height is 95 cm. Find the height of the water left in the underground tank after the overhead tank has been completely filled with water from the underground tank which had been full. Compare the capacity of the overhead tank with that of the underground tank. [Use \( \pi = 3.14 \)]
Answer: Height of water left = 47.5 cm; Capacity of tank is half the capacity of sump.