NCERT Solutions: Surface Areas & Volumes (Exercise 12.1)

# Surface Areas & Volumes (Exercise 12.1) NCERT Solutions - Mathematics (Maths) Class 10

Q1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
Sol.  Volume of each cube = 64 cm3
∴ Total volume of the two cubes = 2 × 64 cm3 = 128 cm3
Let the edge of each cube = x
∴ x3 = 64 = 43
⇒ x =4 cm
Now, the Length of the resulting cuboid 'l' = 2x cm = 8cm
Breadth of the resulting cuboid 'b' = x cm = 4 cm
Height of the resulting cuboid 'h' = x cm = 4 cm
∴ Surface area of the cuboid = 2 (lb + bh + hl)
= 2 [(8 × 4) + (4 × 4) + (4 × 8)]
= 2 [32 + 16 + 32] cm2 = 2 [80] cm2 = 160 cm2.

Q2. 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.
Sol.  For the cylindrical part
Height (h) = 6 cm
∴ Curved surface area
= 2πrh

For hemispherical part
∴ Surface area = 2πr2

∴ Total surface area = CSA of cylinder + CSA of hemisphere
= (264 + 308) cm2 = 572 cm2.

Q3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Sol. Here, r = 3.5 cm
∴ h = (15.5 − 3.5) cm = 12.0 cm
Surface area of the conical part = πrl
Surface area of the hemispherical part = 2πr2

Q4. 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.
Sol. Side of the block = 7 cm
⇒ The greatest diameter of the hemisphere = 7 cm
Surface area of the solid:
= [Total S.A. of the cubical block] + [S.A. of the hemisphere] − [Base area of the hemisphere]
= (6 × l2) + 2πr2 − πr2

Q5. 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.
Sol. Let ‘l’ be the side of the cube.
∴ The greatest diameter of the curved hemisphere = l
⇒ Radius of the curved hemisphere
∴ Surface area of hemisphere = 2πr2

Base area of the hemisphere
Surface area of the cube = 6 × l2 = 6l2
∴ Surface area of the remaining solid

Q6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig.). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Sol. Radius of the hemispherical part
= 5/2 mm = 2.5 mm
∴ Surface area of one hemispherical part = 2πr2
⇒ Surface area of both hemispherical parts

Area of cylindrical part =
∴ Total surface area

Q7. 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 Rs 500 per m2. (Note that the base of the tent will not be covered with canvas.)
Sol.

For cylindrical part:
Radius (r) = 4/2 m = 2 m
Height (h) = 2.1 m
∴ Curved surface area = 2πrh =
For conical part:
Slant height (l) = 2.8 m
∴ Curved surface area
Total surface area
= [Surface area of the cylindrical part] + [Surface area of conical part]

Cost of the canvas used
Cost of 1 m2 of canvas = Rs 500
∴ Cost of 44 m2 of canvas = Rs 500 × 44 = Rs. 22000

Q8. 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.
Sol.

For cylindrical part:
Height = 2.4 cm
Diameter = 1.4 cm
⇒ Radius (r) = 0.7 cm
For conical part:
Base area (r) = 0.7 cm
Height (h) = 2.4 cm
∴ Slant height
∴ Curved surface area of the conical part

Base area of the conical part

Total surface area of the remaining solid:

Q9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 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.

Sol. Radius of the cylinder (r) = 3.5 cm
Height of the cylinder (h) = 10 cm
∴ Total surface area = 2πrh + 2πr2 = 2πr (h + r)

Curved surface area of a hemisphere = 2πr2
Curved surface area of both hemispheres

Base area of a hemisphere = πr2

∴ Base area of both hemispheres = 2πr2

∴ Total surface area of the remaining solid
= 297 cm2 + 154 cm2 − 77 cm2
= (451 − 77) cm2 = 374 cm2.

The document Surface Areas & Volumes (Exercise 12.1) NCERT Solutions | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## Mathematics (Maths) Class 10

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