Q1.
A cuboid has a length of 10 cm, a breadth of 8 cm, and a height of 6 cm. What is its volume?
- 480 cm³
- 240 cm³
- 360 cm³
- 520 cm³
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Answer: A. 480 cm³
The volume of a cuboid is given by the formula V = length × breadth × height. So, V = 10 cm × 8 cm × 6 cm = 480 cm³.
Q2.
Calculate the volume of a cylinder with a radius of 7 cm and a height of 15 cm. (Use π = 22/7)
- 2210 cm³
- 2310 cm³
- 1155 cm³
- 3080 cm³
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Answer: B. 2310 cm³
The volume of a cylinder is given by the formula V = πr²h. Given r = 7 cm, h = 15 cm. V = (22/7) × 7² × 15 = (22/7) × 49 × 15 = 22 × 7 × 15 = 154 × 15 = 2310 cm³.
Q3.
What is the volume of a cube whose side length is 8 cm?
- 64 cm³
- 256 cm³
- 512 cm³
- 128 cm³
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Answer: C. 512 cm³
The volume of a cube is given by the formula V = a³, where 'a' is the side length. Given a = 8 cm. V = 8³ = 8 × 8 × 8 = 512 cm³.
Q4.
The bar chart shows the volumes of different 3D shapes. Which shape has the largest volume?
- Cuboid
- Cube
- Cylinder
- Sphere
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Answer: C. Cylinder
From the bar chart data: Cuboid = 480 cm³, Cube = 512 cm³, Cylinder = 2310 cm³, Cone = 770 cm³, Sphere = 1437 cm³. Comparing these values, the Cylinder has the largest volume (2310 cm³).
Q5.
Find the volume of a cone with a radius of 7 cm and a height of 24 cm. (Use π = 22/7)
- 1232 cm³
- 1540 cm³
- 1848 cm³
- 2464 cm³
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Answer: A. 1232 cm³
The volume of a cone is given by the formula V = (1/3)πr²h. Given r = 7 cm, h = 24 cm. V = (1/3) × (22/7) × 7² × 24 = (1/3) × (22/7) × 49 × 24 = 22 × 7 × 8 = 154 × 8 = 1232 cm³.
Q6.
What is the ratio of the volumes of a cylinder, a cone, and a sphere, if they all have the same radius 'r' and the height of the cylinder and cone is equal to the diameter of the sphere (i.e., h = 2r)?
- 1:2:3
- 3:1:2
- 2:1:3
- 3:2:1
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Answer: B. 3:1:2
Let the radius be 'r' and height 'h'. Given h = 2r.
Volume of cylinder (V_cyl) = πr²h = πr²(2r) = 2πr³
Volume of cone (V_cone) = (1/3)πr²h = (1/3)πr²(2r) = (2/3)πr³
Volume of sphere (V_sph) = (4/3)πr³
Ratio V_cyl : V_cone : V_sph = 2πr³ : (2/3)πr³ : (4/3)πr³
Multiplying by 3/(πr³) to simplify: 6 : 2 : 4 = 3 : 1 : 2.
Q7.
A hemisphere has a radius of 3 cm. What is its volume?
- 36π cm³
- 54π cm³
- 18π cm³
- 27π cm³
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Answer: C. 18π cm³
The volume of a hemisphere is given by the formula V = (2/3)πr³. Given r = 3 cm. V = (2/3)π(3)³ = (2/3)π × 27 = 2 × 9π = 18π cm³.
Q8.
The line chart illustrates how the volume of a sphere changes with its radius. Based on the chart, approximately how many times does the volume increase when the radius doubles from 2 cm to 4 cm?
- About 8 times
- About 4 times
- About 2 times
- About 16 times
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Answer: A. About 8 times
From the line chart: Volume at r=2 cm is approximately 33.5 cm³. Volume at r=4 cm is approximately 268 cm³. The increase factor is 268 / 33.5 ≈ 8. This demonstrates that the volume of a sphere is proportional to the cube of its radius (V ∝ r³), so doubling the radius increases the volume by 2³ = 8 times.
Q9.
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. If the total length of the capsule is 12.5 mm and its diameter is 3.5 mm, what is its total volume? (Use π = 22/7)
- 109.08 mm³
- 105.5 mm³
- 112.3 mm³
- 115.7 mm³
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Answer: A. 109.08 mm³
Radius (r) = Diameter/2 = 3.5/2 = 1.75 mm.
Length of cylindrical part (h_cyl) = Total length - 2 × radius = 12.5 - 2 × 1.75 = 12.5 - 3.5 = 9 mm.
Volume of capsule = Volume of cylinder + Volume of two hemispheres (which is a sphere)
V = πr²h_cyl + (4/3)πr³
V = (22/7) × (1.75)² × 9 + (4/3) × (22/7) × (1.75)³
V = (22/7) × 3.0625 × 9 + (4/3) × (22/7) × 5.359375
V = 22 × 0.4375 × 9 + (4/3) × 22 × 0.765625
V = 86.625 + 22.4583 ≈ 109.0833 mm³.
Q10.
A grain silo is shaped like a cylinder surmounted by a conical top. The cylindrical part has a radius of 7 meters and a height of 10 meters. The conical part has the same radius and a height of 6 meters. Find the total volume of the silo. (Use π = 22/7)
- 1540 m³
- 1848 m³
- 2156 m³
- 2464 m³
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Answer: B. 1848 m³
Volume of cylindrical part (V_cyl) = πr²h_cyl = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 m³.
Volume of conical part (V_cone) = (1/3)πr²h_cone = (1/3) × (22/7) × 7² × 6 = (1/3) × (22/7) × 49 × 6 = 22 × 7 × 2 = 308 m³.
Total volume of silo = V_cyl + V_cone = 1540 + 308 = 1848 m³.
Q11.
A well with an inner diameter of 3 meters is dug 14 meters deep. The earth taken out of it has been evenly spread all around it to a width of 4 meters to form an embankment. Find the height of the embankment.
- 1.0 m
- 1.25 m
- 1.125 m
- 1.5 m
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Answer: C. 1.125 m
Radius of the well (r_well) = 3/2 = 1.5 m. Depth of the well (h_well) = 14 m.
Volume of earth dug out (V_earth) = πr_well²h_well = π(1.5)²(14) = π(2.25)(14) = 31.5π m³.
The earth is spread to form an embankment with a width of 4 m.
Inner radius of embankment (R_inner) = r_well = 1.5 m.
Outer radius of embankment (R_outer) = R_inner + width = 1.5 + 4 = 5.5 m.
Volume of embankment (V_embankment) = π(R_outer² - R_inner²)h_embankment
V_embankment = π(5.5² - 1.5²)h_embankment = π(30.25 - 2.25)h_embankment = π(28)h_embankment.
Since V_earth = V_embankment:
31.5π = 28π × h_embankment
h_embankment = 31.5 / 28 = 1.125 m.
Q12.
A combined solid is made up of a cylindrical part, a conical part, and a hemispherical part. The pie chart shows the percentage distribution of their volumes. If the total volume of the combined solid is 2000 cm³, what is the volume of the conical part?
- 1100 cm³
- 500 cm³
- 400 cm³
- 250 cm³
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Answer: B. 500 cm³
From the pie chart, the conical part constitutes 25% of the total volume.
Total volume = 2000 cm³.
Volume of conical part = 25% of 2000 cm³ = (25/100) × 2000 = 500 cm³.
Q13.
An iron sphere of radius 12 cm is melted and recast into smaller spherical balls, each of radius 2 cm. How many such small spherical balls can be formed?
- 36
- 216
- 72
- 144
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Answer: B. 216
Volume of the large sphere (V_large) = (4/3)πR³, where R = 12 cm.
Volume of each small sphere (V_small) = (4/3)πr³, where r = 2 cm.
Number of small spheres = V_large / V_small = [(4/3)π(12)³] / [(4/3)π(2)³] = (12/2)³ = 6³ = 216.
Q14.
A solid metallic cone with a radius of 6 cm and a height of 3 cm is melted and recast into a solid sphere. What is the radius of the sphere?
- 2 cm
- 2.5 cm
- 3 cm
- 3.5 cm
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Answer: C. 3 cm
Volume of the cone (V_cone) = (1/3)πr²h. Given r = 6 cm, h = 3 cm.
V_cone = (1/3)π(6)²(3) = (1/3)π(36)(3) = 36π cm³.
When the cone is melted and recast into a sphere, the volume remains the same.
Volume of the sphere (V_sphere) = (4/3)πR³, where R is the radius of the sphere.
Equating the volumes: (4/3)πR³ = 36π
R³ = 36 × (3/4) = 9 × 3 = 27
R = ³√27 = 3 cm.
Q15.
A solid cuboid of dimensions 6 cm × 4 cm × 2 cm is melted and drawn into a wire of uniform diameter 2 mm. Find the length of the wire. (Use π = 22/7)
- 15.27 m
- 12.50 m
- 18.75 m
- 20.00 m
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Answer: A. 15.27 m
Volume of the cuboid (V_cuboid) = length × breadth × height = 6 cm × 4 cm × 2 cm = 48 cm³.
The wire is cylindrical. Diameter of wire = 2 mm = 0.2 cm.
Radius of wire (r) = 0.2/2 = 0.1 cm.
Let the length of the wire be L.
Volume of the wire (V_wire) = πr²L = π(0.1)²L = 0.01πL cm³.
Since the cuboid is melted and recast into a wire, their volumes are equal:
V_cuboid = V_wire
48 = 0.01πL
L = 48 / (0.01π) = 4800/π
Using π = 22/7: L = 4800 × 7 / 22 = 2400 × 7 / 11 = 16800 / 11 ≈ 1527.27 cm.
Converting to meters: L = 1527.27 / 100 = 15.27 meters.
