Volume of Sphere

Exercise 13.8

Question 1: Find the volume of a sphere whose radius is
(i) 7 cm

Answer: Volume of sphere `= 4/3 \π r^3`
`= (4/3) xx (22/7) xx 7^3 = 1437.33` cubic cm

(ii) 0.63 m

Answer: Volume of sphere `= 4/3\ π r^3`
`= (4/3) xx (22/7) xx (0.63)^3 = 1.047` cubic m

Question 2: Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm

Answer: Volume of sphere `= 4/3\ π r^3`
`= (4/3) xx (22/7) xx 28^3 = 91989.33` cubic cm

(ii) 0.21 m

Answer: Volume of sphere `= 4/3\ π r^3`
`= (4/3) xx (22/7) xx (0.21)^3 = 0.38808` cubic m


Question 3: The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3?

Answer: Volume of sphere `= 4/3\ π r^3`
`= (4/3) xx (22/7) xx (4.2)^3 = 310.464` cubic cm

Mass = volume X density
`= 310.464 xx 8.9= 2763.1296  gm = 2.76  kg`

Question 4: The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Answer: Volume of two similar shapes are in triplicate ratio of their dimensions. For example; if radii are R and r then ratio of volumes = R3 : r3
Hence, volume of earth/volume of moon
= 43 : 13
= 64 : 1

Question 5: How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold?

Answer: Volume of hemisphere `= 2/3\ π r^3`
`= (2/3) xx (22/7) xx (5.25)^3 = 303.1875` cubic cm = 0.303 litre


Question 6: A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.

Answer: Inner radius r = 1 m, outer radius r = 1.01 m
Volume of metal `= 4/3\ π (R^3 - r^3)`
`= (4/3) xx (22/7) [(1.01)^3 - 1^3]`
`= (4/3) xx (22/7) xx 0.030301 = 0.06348` cubic m

Question 7: Find the volume of a sphere whose surface area is 154 cm2.

Answer: Surface area of sphere `= 4 π r^2`
Or, `154 = 4 xx (22/7) xx r^2`
Or, `r^2 = (154 xx 7)/(22 xx 4) = 49/4`
Or, `r = 7/2 = 3.5` cm

Volume of sphere `= 4/3 \π r^3`
`= (4/3) xx (22/7) xx (3.5)^3 = 179.67` cubic cm

Question 8: A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs 498.96. If the cost of white-washing is Rs 2.00 per square metre, find the
(i) inside surface area of the dome, (ii) volume of the air inside the dome.

Answer: Curved surface area of hemisphere = cost/rate
`= 498.96/2 = 249.48` sq m
Or, `2 π r^2 = 249.48`
Or, `r^2 = (249.48 xx 7)/(2 xx 22)`
Or, `r = 6.3` m

Volume of hemisphere `= 2/3\ π r^3`
`= (2/3) xx (22/7) xx (6.3)^3 = 523.908` cubic m

Question 9: Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. Find the
(i) radius r′ of the new sphere, (ii) ratio of S and S′.

Answer: Here; ratio of volumes = 27 : 1
Radii shall be in sub-triplicate ratio, i.e. 3 : 1
Because 33 : 13 = 27 : 1
Now, surface areas shall be in duplicate ratio of radii
Hence, ratio of surface areas = 32 : 12 = 9 : 1

Question 10: A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to fill this capsule?

Asnwer: Volume of sphere `= 4/3 \π r^3`
`= (4/3) xx (22/7) xx (1.75)^3= 22.46` cubic mm



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