Question 1: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.

**Solution:** radius = 1 cm, height = 1 cm

Volume of hemisphere

`=(2)/(3) πr^3`

`=(2)/(3) πxx1^3`

`=(2)/(3) π cm^3`

Volume of cone

`=(1)/(3) πr^2h`

`=(1)/(3) πxx1^2xx1`

`=(1)/(3) π cm^3`

Total volume

`=(2)/(3) π+(1)/(3) π=π cm^3`

Question 2: Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a 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 that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)

**Solution:** Height of cylinder = 12 – 4 = 8 cm, radius = 1.5 cm, height of cone = 2 cm

Volume of cylinder

`=πr^2h`

`=πxx1.5^2xx8=18π cm^3`

Volume of cone

`=(1)/(3) πr^2h`

`=(1)/(3) πxx1.5^2xx2=1.5π cm^3`

Total volume

`=1.5π+1.5π+18π=21π=66 cm^3`

Question 3: A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.

**Solution:** Length of cylinder = 5 – 2.8 = 2.2 cm, radius = 1.4 cm

Volume of cylinder

`=πr^2h`

`=πxx1.4^2xx2.2`

`=4.312π cm^3`

Volume of two hemispheres

`=(4)/(3) πr^3`

`=(4)/(3) πxx1.4^3`

`=(10.976)/(3) π cm^3`

Total volume

`=4.312π+(10.976)/(3) π`

Volume of syrup = 30% of total volume

`= π(4.312π+(10.976)/(3) π)xx30%`

`=(23.912)/(3)xx(30)/(100)xx(22)/(7)`

`=7.515 cm^3`

Volume of syrup in 45 gulabjamuns `= 45 xx 7.515 = 338.184` cm^{3}

Question 4: A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.

**Solution:** Dimensions of cuboid = 15 cm x 10 cm x 3.5 cm, radius of cone = 0.5 cm, depth of cone = 1.4 cm

Volume of cuboid = length x width x height

= 15 x 10 x 3.5 = 525 cm^{3}

Volume of cone

`=(1)/(3) πr^2h`

`=(1)/(3)xx(22)/(7)xx0.5^2xx1.4`

`=(11)/(30) cm^3`

Volume of wood = Volume of cuboid – 6 x volume of cone

`=525-6xx(11)/(3)`

`=525-(11)/(5)=522.8 cm^3`

Question 5: A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

**Solution:** radius of cone = 5 cm, height of cone = 8 cm, radius of sphere = 0.5 cm

Volume of cone

`=(1)/(3) πr^2h`

`=(1)/(3) πxx5^2xx8`

`=(200)/(3) π cm^3`

Volume of lead shot

`=(4)/(3) πr^3`

`=(4)/(3) πxx0.5^3`

`=(1)/(6) π cm^3`

Number of lead shots

`=(200)/(3) πxx(1)/(4)÷(1)/(6) π`

`=(50π)/(3)xx(6)/(π)=100`

Question 6: A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8g mass.

**Solution:** radius of bigger cylinder = 12 cm, height of bigger cylinder = 220 cm

Radius of smaller cylinder = 8 cm, height of smaller cylinder = 60 cm

Volume of bigger cylinder

`= πr^2h`

`=πxx12^2xx220`

`=31680π cm^3`

Volume of smaller cylinder

`=πr^2h`

`=πxx8^2xx60`

`=3840π cm^3`

Total volume

`=31680π+3840π=35520π cm^3`

Mass = Density x volume

`=8xx35520π`

`=892262.4 gm=892.3 kg`

Question 7: A solid consisting of a right circular cone of height 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 such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

**Solution:** Radius of cone = 60 cm, height of cone = 120 cm

Radius of hemisphere = 60 cm

Radius of cylinder = 60 cm, height of cylinder = 180 cm

Volume of cone

`= (1)/(3)πr^2h`

`=(1)/(3) πxx60^2xx120`

`=144000π cm^3`

Volume of hemisphere

`=(2)/(3) πr^3`

`=(2)/(3) πxx60^3`

`=144000π cm^3`

Volume of solid

`=(144000+144000) π=288000π cm^3`

Volume of cylinder

`=πr^2h`

`=πxx60^2xx180`

`=648000π cm^3`

Volume of water left in the cylinder

`=(648000+288000) π=1130400 cm^3`

Question 8: A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm^{3}. Check whether she is correct, taking the above as the inside measurements, and π = 3.14.

**Solution:** Radius of cylinder = 1 cm, height of cylinder = 8 cm, radius of sphere = 8.5 cm

Volume of cylinder

`= πr^2h`

`=πxx1^2xx8`

`=8π cm^3`

Volume of sphere

`=(4)/(3) πr^3`

`=(4)/(3) πxx((8.5)/(2))^3`

`=(614125)/(6000) π cm^3`

Total volume

`=((614125)/(6000)+8) π`

`=((614125+48000)/(6000)) π`

`=346.51 cm^3`

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