Unless stated otherwise, take
LetÂ theÂ lengthÂ of eachÂ edge ofÂ the cubeÂ be a cm.
Then, VolumeÂ
Â
Â Â a = 4
When two cubes of equal volumes (i.e., equal edges are joined end to end, we get a cuboid such that its.
l = Length = 4cm + 4cm = 8 cm
b = Breadth = 4 cm
and Â Â Â h = Height = 4 cm
ThereforeÂ Surface area of the cuboid
= 2(lb + bh + hl)
= 2(8 Ã— 4 + 4 Ã— 4 + 4 Ã— 8)
= 2(32 + 16 + 32)
= (2 Ã— 80) = 160
Q.2Â A vessel is in the form of a hollowÂ hemisphere mounted by a hollow cylinder. The diameter of theÂ hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Sol.Â
Here r, theÂ radius of hemisphereÂ = 7 cm ,
h , the heightÂ of cylinder = (13 â€“ 7) cm = 6 cm
Clearly, radius of the base of cylindrical part is also r cm .
Surface area of the vessel = Curved surface area of the cylindrical part + Curved surface area of hemispherical part
Q.3Â A toy is in theÂ form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The totalÂ height of the toy is 15.5 cm. FindÂ the total surface area of the toy.
Sol.
We have VO = 15.5 cm,
OA = OO' = 3.5 cm
Let r be theÂ radius of the base of cone and h be the height of conical part of the by.
Then r = OA = 3.5 cm
h = VO = VO' â€“ OO = (15.5 â€“ 3.5) cm = 12 cm
AlsoÂ radiusÂ of the hemisphere = OA = r = 3.5 cm
Total surface area of the toy
= Curved surface area of cone + Curved surface area of hemisphere
where l = Slant height
TheÂ greatest diameter that a hemisphereÂ can have = 7 cm .
Surface area of the solidÂ after surmounted hemisphere
= 294 + 38.5Â = 332.5
Q.5Â A hemispherical depression is cutÂ ofÂ one face of a cubical wooden blockÂ such theÂ diameter l of theÂ hemisphere is equal to the edge of the cube. Determine th surface area of theÂ remaining solid.
Sol.
Edge of the cubeÂ =
Diameter of the hemisphereÂ =
ThereforeÂ RadiusÂ of the hemisphere =
ThereforeÂ AreaÂ of the remainingÂ solidÂ after cutting out theÂ hemisphericalÂ depression.
Q.6Â A medicine capsule is in theÂ shape of a cylinder with two hemispheres stuck to each of its ends (see figure). The length of the entire capsule is 14 mm and theÂ diameter of theÂ capsule is 5 mm. FindÂ its surface area.
Sol.
Let r cmÂ be theÂ radius and h cmÂ be the heightÂ of theÂ cylinder. Then.
and Â Â Â
= (14 â€“ 5)mm = 9mm
Also theÂ radius of hemisphere =
Now, surface area of the capsule = Curved surface of cylinder + Surface area of two hemispheres
Q.7 A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of theÂ cylindrical part are 2.1 m and 4 m respectively, and theÂ slant heightÂ of the top is 2.8 m, findÂ the area of the canvas used for making the tent. Also, findÂ the cost of the canvas of the tent at the rate of Rs 500 per . (Note that the base of the tent will not covered withÂ canvas).
Sol.Â
We have, TotalÂ canvas usedÂ =Â Curved surface area of cylinder + Curved surface areaÂ of cone
Now,Â cost of the canvas for theÂ tent = Rs 500
So,Â cost of theÂ canvas for theÂ tent = Rs 44 Ã— 500 = Rs 22000
Q.8 FromÂ a solidÂ cylinder whose heightÂ is 2.4 cm and diameter 1.4 cm, a conicalÂ cavity of the same heightÂ and same diameter isÂ hollowed out. Find theÂ total surface area of the remainingÂ solid to theÂ nearest .
Sol.
RadiusÂ of the cylinder
Height of the cylinder = 2.4 cm
Radius of the cone = .7 cm
HeightÂ of the cone = 2.4 cm
SlantÂ heightÂ of the cone
Surface area of the remainingÂ solid = Curved surface area of cylinder + Curved surface of theÂ cone + Area of upper circular base of cylinder
Q.9Â A wooden articleÂ was made by scooping outÂ a hemisphereÂ from eachÂ end of a solid cylinder, as shown in figure. If the heightÂ of the cylinder is 10 cm,and its base is of radiusÂ 3.5 m, find theÂ totalÂ surface area of the article.
Sol.
Surface area of the article when it is ready = Curved surface area of cylinder + 2 Ã— Curved surface area of hemisphere.
whereÂ r = 3.5 m and h = 10 cm