Unless stated otherwise, take .
Volume of theÂ solid = VolumeÂ of the coneÂ + VolumeÂ of theÂ hemisphere
Â Â Â Â Â Â [Since h = r and R = r]
Â Â Â [Since r = 1 cm]
Q.2Â Â Â Â Rachel an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends by using thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm,find the volume of air contained in the model the Rachel made. (Assume the outer and inner dimensions of the model be nearly the same).
Sol.
VolumeÂ of the airÂ contained in the model =Â Volume of theÂ cylindrical portion of the model + VolumeÂ of its twoÂ conical ends.
where
Q.3Â Â Â Â Â Â A gulab jamun, contains sugar syrup up toÂ about 30% of its volume. Find approximately how muchÂ syrup would be found in 45 gulab jamuns,Â each shaped like a cylinder with two hemsipherical ends with length 5 cm and diameter is 2.8 cm (see figure).
Sol.
Volume of the gulab jamun = Volume of the cylindrical portion + Volume of the hemispherical ends
, whereÂ rÂ = 1.4 cm, h = 2.2 cm
VolumeÂ of 45 gulab jamuns
Quantity of syrup in gulab jamunsÂ = 30% of theirÂ volume
Q.4Â Â Â Â Â A penÂ standÂ made of woodÂ is in theÂ shapeÂ of a cuboidÂ with fourÂ conicalÂ depressions to hold pens. TheÂ dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 m and theÂ depth is 1.4 cm. Find the volumeÂ of woodÂ in theÂ entireÂ stand (see figure).
Sol.
VolumeÂ of woodÂ in the entireÂ standÂ Â = VolumeÂ of the cuboid â€“ 4 Ã— VolumeÂ of a depression (i.e., cone)
Q.5Â Â Â Â Â Â A vessel in the form of an inverted cone. Its height is 8 cm and the radius of its top,Â which is open, is 5 cm. It is filled with water up to the brim. When lead shots, eachÂ of which is a sphere of radius 0.5 cm are dropped intoÂ the vessel, one- fourth of theÂ water flows out. Find theÂ number of lead shots dropped in the vessel.
Sol.Â
HeightÂ of the conicalÂ vessel, h = 8 cm.
ItsÂ radius r = 5 cm
Volume of cone = Volume of water in cone
VolumeÂ of water flows out = Volume ofÂ lead shots
of theÂ volume of water in the cone
RadiusÂ of the lead shotÂ = 0.5 cm
Volume of oneÂ sphericalÂ lead shot
ThereforeÂ Â Number of leadÂ shots dropped into the vessel
Q.6Â Â Â Â Â Â A solid ironÂ poleÂ consists of a cylindrical heightÂ 220 cm and base diameter 24 cm, whichÂ is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 of ironÂ has approximately 8 g mass. (Use )
Sol.
VolumeÂ of the solid ironÂ pole = VolumeÂ of the cylindricalÂ portion + VolumeÂ of the otherÂ cylindrical portion
Hence, the mass of the pole = (111532.8 Ã— 8) grams
Q.7Â Â Â Â Â A solidÂ consistingÂ of a rightÂ circularÂ cone of heightÂ 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water circular such that it touches the bottom. Find the volume of water left in the cylinder, if theÂ radius of theÂ cylinder is 60 cm and its height is 180 cm.
Sol.
VolumeÂ of the cylinder
Volume of theÂ solid = Volume of cone + Volume of hemisphere
Volume of water left in the cylinder = VolumeÂ of the cylinder â€“ VolumeÂ of the solid
Volume of sphericalÂ vessel
= 3.14[8 + 102.353]
= 3.14 Ã— 110.353Â = 346.51
ThereforeÂ HereÂ answer is incorrect.
Correct answer is