Exercise 12.1 Solutions
Surface Areas and Volumes
(Unless stated otherwise, take π = 22/7)
1. 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Step 1: Find the side of each cube.
Volume of a cube = (side)³ = a³.
Given, a³ = 64 cm³. So, a = ∛64 = 4 cm.
Step 2: Determine the dimensions of the resulting cuboid.
When two cubes are joined end to end, the length doubles while the breadth and height remain the same.
Length (l) = 4 + 4 = 8 cm.
Breadth (b) = 4 cm.
Height (h) = 4 cm.
Step 3: Calculate the surface area of the cuboid.
Surface Area = 2(lb + bh + hl)
= 2( (8 × 4) + (4 × 4) + (4 × 8) )
= 2(32 + 16 + 32) = 2(80) = 160 cm².
Answer: The surface area of the resulting cuboid is 160 cm².
2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Step 1: Identify dimensions.
Diameter of hemisphere = 14 cm, so radius (r) = 7 cm.
Total height = 13 cm.
Height of the cylindrical part (h) = Total height – radius of hemisphere = 13 – 7 = 6 cm.
Step 2: Determine the required surface area.
Inner surface area of vessel = (Inner CSA of cylinder) + (Inner CSA of hemisphere).
CSA of cylinder = 2πrh
CSA of hemisphere = 2πr²
Step 3: Calculate the total inner surface area.
Total Area = 2πrh + 2πr² = 2πr(h + r)
= 2 × (22/7) × 7 × (6 + 7)
= 44 × 13 = 572 cm².
Answer: The inner surface area of the vessel is 572 cm².
3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Step 1: Identify dimensions.
Radius (r) = 3.5 cm.
Total height = 15.5 cm.
Height of cone (h) = Total height – radius of hemisphere = 15.5 – 3.5 = 12 cm.
Step 2: Calculate the slant height (l) of the cone.
l = √(r² + h²) = √((3.5)² + (12)²) = √(12.25 + 144) = √156.25 = 12.5 cm.
Step 3: Calculate the total surface area.
Total Surface Area = (CSA of cone) + (CSA of hemisphere)
= πrl + 2πr² = πr(l + 2r)
= (22/7) × 3.5 × (12.5 + 2 × 3.5)
= 11 × (12.5 + 7) = 11 × 19.5 = 214.5 cm².
Answer: The total surface area of the toy is 214.5 cm².
4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
Step 1: Determine the greatest diameter.
The greatest diameter the hemisphere can have is equal to the side of the cube.
Greatest diameter = 7 cm. So, radius (r) = 3.5 cm.
Step 2: Determine the formula for the surface area.
Surface Area = (TSA of cube) – (Area of base of hemisphere) + (CSA of hemisphere)
= 6a² - πr² + 2πr² = 6a² + πr²
Step 3: Calculate the surface area.
= 6(7)² + (22/7)(3.5)²
= 6(49) + (22/7)(12.25) = 294 + 38.5 = 332.5 cm².
Answer: The surface area of the solid is 332.5 cm².
5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Step 1: Identify dimensions.
Edge of cube (a) = Diameter of hemisphere = l.
Radius of hemisphere (r) = l/2.
Step 2: Determine the formula for the surface area.
The logic is the same as in Q4. The area of the top face of the cube is covered by the circle, but the inner curved surface of the hemisphere is now exposed.
Surface Area = (TSA of cube) – (Area of circular top) + (Inner CSA of hemisphere)
= 6a² - πr² + 2πr² = 6a² + πr²
Step 3: Substitute the given variables.
= 6(l)² + π(l/2)² = 6l² + (πl²)/4
We can factor out l²: l²(6 + π/4) or l²4(24 + π).
Answer: The surface area of the remaining solid is 6l² + (πl²)/4.
6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
>Step 1: Identify dimensions.
Diameter = 5 mm, so radius (r) = 2.5 mm.
Total length = 14 mm.
Length of cylindrical part (h) = Total length – (2 × radius of hemisphere) = 14 – (2 × 2.5) = 14 – 5 = 9 mm.
Step 2: Determine the formula for the surface area.
Surface Area = (CSA of cylinder) + (CSA of two hemispheres)
= 2πrh + 2(2πr²) = 2πr(h + 2r)
Step 3: Calculate the surface area.
= 2 × (22/7) × 2.5 × (9 + 2 × 2.5)
= 2 × (22/7) × 2.5 × (14)
= 2 × 22 × 2.5 × 2 = 220 mm².
Answer: The surface area of the capsule is 220 mm².
7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500 per m².
Step 1: Identify dimensions.
Cylinder height (h) = 2.1 m, Diameter = 4 m, Radius (r) = 2 m.
Cone slant height (l) = 2.8 m.
Step 2: Calculate the area of the canvas.
Area of Canvas = (CSA of cylinder) + (CSA of cone)
= 2πrh + πrl = πr(2h + l)
= (22/7) × 2 × (2 × 2.1 + 2.8)
= (44/7) × (4.2 + 2.8) = (44/7) × 7 = 44 m².
Step 3: Calculate the cost.
Cost = Area × Rate = 44 × 500 = ₹22000.
Answer: Area of canvas is 44 m² and the cost is ₹22,000.
8. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².
Step 1: Identify dimensions.
Height (h) = 2.4 cm, Diameter = 1.4 cm, Radius (r) = 0.7 cm.
Step 2: Calculate the slant height (l) of the conical cavity.
l = √(r² + h²) = √((0.7)² + (2.4)²) = √(0.49 + 5.76) = √6.25 = 2.5 cm.
Step 3: Determine the formula for the surface area.
Total Surface Area = (CSA of cylinder) + (Area of base) + (Inner CSA of cone)
= 2πrh + πr² + πrl = πr(2h + r + l)
Step 4: Calculate the surface area.
= (22/7) × 0.7 × (2 × 2.4 + 0.7 + 2.5)
= 2.2 × (4.8 + 0.7 + 2.5) = 2.2 × 8.0 = 17.6 cm².
Answer: The total surface area of the remaining solid is approximately 18 cm².
9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.
>Step 1: Identify dimensions.
Height of cylinder (h) = 10 cm.
Radius (r) = 3.5 cm.
Step 2: Determine the formula for the surface area.
Total Surface Area = (CSA of cylinder) + 2 × (Inner CSA of a hemisphere)
= 2πrh + 2(2πr²) = 2πr(h + 2r)
Step 3: Calculate the surface area.
= 2 × (22/7) × 3.5 × (10 + 2 × 3.5)
= 22 × (10 + 7) = 22 × 17 = 374 cm².
Answer: The total surface area of the article is 374 cm².
