Volume of Sphere

Exercise 13.8

Question 1: Find the volume of a sphere whose radius is

(i) 7 cm

Answer: Volume of sphere `=4/3πr^3`

`=4/3xx(22)/7xx7^3=1437.33` cubic cm

(ii) 0.63 m

Answer: Volume of sphere `=4/3πr^3`

`=4/3xx(22)/7xx0.63^3=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 `=4/3πr^3`

`=4/3xx(22)/7xx28^3=91989.33` cubic cm

(ii) 0.21 m

Answer: Volume of sphere `=4/3πr^3`

`=4/3xx(22)/7xx0.21^3=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 `=4/3πr^3`

`=4/3xx(22)/7xx4.2^3=310.464` cubic cm

Mass = Volume × Density

`=310.464xx8.9=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 `=2/3πr^3`

`=2/3xx(22)/7xx5.25^3=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 `=4/3π(R^3-r^3)`

`=4/3xx(22)/7(1.01^3 - 1^3)`

`=4/3xx(22)/7xx0.030301=0.06348` cubic m

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

Answer: Surface area of sphere `=4πr^2`

Or, `154=4xx(22)/7xx\r^2`

Or, `r^2=(154xx7)/(22xx4)=(49)/4`

Or, `r=7/2=3.5` cm

Volume of sphere `=4/3πr^3`

`=4/3xx(22)/7xx3.5^3=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, `2πr^2=249.48`

Or, `r^2=(249.48xx7)/(2xx22)`

Or, `r=6.3` m

Volume of hemisphere `=2/3πr^3`

`=2/3xx(22)/7xx6.3^3=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 `=4/3πr^3`

`=4/3xx(22)/7xx1.75^3=22.46` cubic mm