Surface Areas and Volumes Ex 13.2 Solutions
Question - 1 : - A solid is in theshape of a cone standing on a hemisphere with both their radii being equal to 1cm and the height of the cone is equal to its radius. Find the volume of thesolid in terms of π.
Answer - 1 : - 
Question - 2 : - Rachel, anengineering student, was asked to make a model shaped like a cylinder with twocones attached at its two ends by using a thin aluminum sheet.
Answer - 2 : - 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.)
Question - 3 : - A Gulab jamun contains sugar syrup up to about30% of its volume. Find approximately how much syrup would be found in 45 Gulabjamuns,
Answer - 3 : - each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see figure).
Question - 4 : - A pen stand made ofwood 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.
Answer - 4 : - 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 (see Fig.).
Solution:
Volume of cuboid = length x width x height
We know the cuboid’s dimensions as 15 cmx10cmx3.5 cm
So, the volume of the cuboid = 15x10x3.5 = 525cm3
Here, depressions are like cones and we know,
Volume of cone = (⅓)πr2h
Given, radius (r) = 0.5 cm and depth (h) = 1.4cm
∴ Volume of 4 cones = 4x(⅓)πr2h
= 1.46 cm2
Now, volume of wood = Volume of cuboid – 4 xvolume of cone
= 525-1.46 = 523.54 cm2
A vessel is in theform of an inverted cone. Its height is 8 cm and the radius of its top, whichis open, is 5 cm. It is filled with water up to the brim. When lead shots,
Answer - 5 : - 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:For the cone,
Radius = 5 cm,
Height = 8 cm
Also,
Radius of sphere = 0.5 cm
The diagram will be like
It is known that,
Volume of cone = volume of water in the cone
= ⅓πr2h = (200/3)π cm3
Now,
Total volume of water overflown= (¼)×(200/3) π=(50/3)π
Volume of lead shot
= (4/3)πr3
= (1/6) π
Now,
The number of lead shots = Total Volume ofWater over flown/ Volume of Lead shot
= (50/3)π/(⅙)π
= (50/3)×6 = 100
Question - 6 : - A solid iron pole consists of a cylinder ofheight 220 cm and base diameter 24 cm, which is surmounted by another cylinderof height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 ofiron has approximately 8 g mass.
Answer - 6 : -
Given, the height of the big cylinder (H) =220 cm
Radius of the base (R) = 24/12 = 12 cm
So, the volume of the big cylinder = πR2H
= π(12)2 × 220 cm3
= 99565.8 cm3
Now, the height of smaller cylinder (h) = 60cm
Radius of the base (r) = 8 cm
So, the volume of the smaller cylinder = πr2h
= π(8)2×60 cm3
= 12068.5 cm3
∴ Volume of iron = Volume of the big cylinder+ Volume of thesmall cylinder
= 99565.8 + 12068.5
=111634.5 cm3
We know,
Mass = Density x volume
So, mass of the pole = 8×111634.5
= 893 Kg (approx.)
Question - 7 : - A solid consisting of a right circular cone ofheight 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm isplaced upright in a right circular cylinder full of water such that it touchesthe bottom. Find the volume of water left in the cylinder, if the radius of thecylinder is 60 cm and its height is 180 cm.
Answer - 7 : - 
Here, the volume of water left will be =Volume of cylinder – Volume of solid
Given,
Radius of cone = 60 cm,
Height of cone = 120 cm
Radius of cylinder = 60 cm
Height of cylinder = 180 cm
Radius of hemisphere = 60 cm
Now,
Total volume of solid = Volume of Cone +Volume of hemisphere
Volume of cone = π×122×103cm3 =144×103π cm3
So, Total volume of solid = 144×103πcm3 -(⅔)×π×103 cm3
Volume of hemisphere = (⅔)×π×103 cm3
Volume of cylinder = π×602×180 =648000 = 648×103 π cm3
Now, volume of water left will be = Volume ofcylinder – Volume of solid
= (648-288) × 103×π = 1.131 m3
Answer - 8 : - 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.