Q1. 2 cubes each of volume 64 cm^{3} are joined end to end. Find the surface area of the resulting cuboid.
Sol. Volume of each cube = 64 cm^{3}
∴ Total volume of the two cubes = 2 × 64 cm^{3} = 128 cm^{3}
Let the edge of each cube = x
∴ x^{3} = 64 = 4^{3}
⇒ 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] cm^{2} = 2 [80] cm^{2} = 160 cm^{2}.
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
Radius (r) = 7 cm
Height (h) = 6 cm
∴ Curved surface area
= 2πrh
For hemispherical part
Radius (r) = 7 cm
∴ Surface area = 2πr^{2}
∴ Total surface area = CSA of cylinder + CSA of hemisphere
= (264 + 308) cm^{2} = 572 cm^{2}.
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 = 2πr^{2}
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 × l^{2}) + 2πr^{2} − πr^{2}
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 = 2πr^{2}
Base area of the hemisphere
Surface area of the cube = 6 × l^{2} = 6l^{2}
∴ 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 = 2πr^{2}
⇒ 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 m^{2}. (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
Base radius (r)= 2m
∴ 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 m^{2} of canvas = Rs 500
∴ Cost of 44 m^{2} 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 cm^{2}.
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 + 2πr^{2} = 2πr (h + r)
Curved surface area of a hemisphere = 2πr^{2}
∴ Curved surface area of both hemispheres
Base area of a hemisphere = πr^{2}
∴ Base area of both hemispheres = 2πr^{2}
∴ Total surface area of the remaining solid
= 297 cm^{2} + 154 cm^{2} − 77 cm^{2}
= (451 − 77) cm^{2} = 374 cm^{2}.
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