Please refer to the following lecture notes for the formulas used in this exercise: Notes

Question 1: A solid sphere of radius 15 \ cm is melted and recast into solid right circular cones of radius 2.5 \ and height 8 \ cm . Calculate the number of cones recast.   [2013]

Answer:

Sphere: Radius = 15 \ cm

Cone: Radius = 2.5 \ cm and Height = 8 \ cm

Therefore number of cones re-casted = \frac{\frac{4}{3} \times \pi \times (15)^3}{\frac{1}{3} \times \pi \times (2.5)^2 \times 8} = 270

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Question 2: A hollow sphere of internal and external diameters 4 \ cm and 8 \ cm respectively is melted into a cone of base diameter 8 \ cm . Find the height of the cone.   [2002]

Answer:

Internal diameter = 4 \ cm \Rightarrow Internal radius = 2 cm

External diameter = 8 \ cm \Rightarrow External radius = 4 cm

Radius of the cone = 4 \ cm

\therefore \ \ \frac{4}{3} \times \pi \times (3)^3 - \frac{4}{3} \times \pi \times (2)^3 = \frac{1}{3} \times \pi \times (4)^2 \times h

\Rightarrow h = \frac{4 \times (64-8)}{16} = 14 \ cm

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Question 3: The radii of the internal and external surfaces of a metallic spherical shell are 3 \ cm and 5 \ cm respectively. It is melted and recast into a solid right circular cone of height 32 \ cm . Find the diameter of the base of the cone.

Answer:

Internal radius = 3 \ cm

External radius = 5 \ cm

Height of the cone = 32 \ cm

\therefore \ \ \frac{4}{3} \times \pi \times (5)^3 - \frac{4}{3} \times \pi \times (5)^3 = \frac{1}{3} \times \pi \times (r)^2 \times 32

\Rightarrow r^2 = \frac{4 \times (125-27)}{32} = 12.25 \ cm

Therefore r = 3.5 \ cm and hence diameter = 7 \ cm

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Question 4: Total volume of three identical cones is the same as that of a bigger cone whose height is 9 \ cm and diameter 40 \ cm . Find the radius of the base of each smaller cone, if height of each is 108 \ cm .

Answer:

Let the radius of the cone = r

Height of the cone = 108 \ cm

Bigger cone: Height = 9 \ cm and radius = 20 \ cm

\therefore  3 \times \frac{1}{3} \times \pi \times (r)^2 \times 108 =  \frac{1}{3} \times \pi \times (20)^2 \times 9

3 r^2 \times 108 = 20^2 \times 9

\Rightarrow r= \frac{20}{6} = 3\frac{1}{3}

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Question 5: A solid rectangular block of metal 49 \ cm by 44 \ cm by 18 \ cm is melted and formed into a solid sphere. Calculate the radius of the sphere.

Answer:

Dimension of the block = 49 \ cm \times 44 \ cm \times 18 \ cm

Let the radius of the sphere = r \ cm

\therefore 49  \times 44  \times 18   = \frac{4}{3} \times \pi \times (r)^3

r^3 = \frac{49  \times 44  \times 18}{4 \times 22} = 21^3 \Rightarrow r = 21 \ cm

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Question 6: A hemispherical bowl of internal radius 9 \ cm is full of liquid. This liquid is to be filled into conical shaped small containers each of diameter 3 \ cm and height 4 \ cm . How many containers are necessary to empty the bowl?

Answer:

Bowl: Internal radius = 9 \ cm

Cone: radius = 1.5 \ cm , Height = 4 \ cm

\therefore  \frac{1}{2} \frac{4}{3} \times \pi \times (9)^3 = n \times \frac{1}{3} \times \pi \times (1.5)^2 \times 4

\Rightarrow n = \frac{9 \times 9 \times 9 }{2 \times 1.5 \times 1.5} = 162

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Question 7: A hemispherical bowl of diameter 7.2 \ cm is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 \ cm . Find the height of the cone if it is completely filled.

Answer:

Hemisphere: Radius = 3.6 \ cm

Cone: Radius = 4.8 \ cm , Height = h

\therefore  \frac{1}{2} \frac{4}{3} \times \pi \times (3.6)^3 =  \frac{1}{3} \times \pi \times (4.8)^2 \times h

\Rightarrow h = \frac{2 \times 3.6^3}{4.8^2} = 4.05 \ cm

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Question 8: A solid cone of radius 5 \ cm and height 8 \ cm is melted and made into small spheres of radius 0.5 \ cm . Find the number of spheres formed.

Answer:

Cone: Radius = 5 \ cm , Height = 8 \ cm

Sphere: Radius = 0.5 \ cm

n \times \frac{4}{3} \times \pi \times (0.5)^3 =  \frac{1}{3} \times \pi \times (5)^2 \times 8

\Rightarrow n = \frac{5^2 \times 8}{4 \times 0.5^3} = 400

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Question 9: The total area of a solid metallic sphere is 1256 \ cm^2 . It is melted and recast into solid right circular cones of radius 2.5 \ cm and height 8 \ cm . Calculate: (i) the radius of the solid sphere, (ii) the number of cones recast.   [2000]

Answer:

Surface area = 1256 \ cm^2

(i) \therefore 4 \pi r^2 = 1256

\Rightarrow r^2 = \frac{1256}{4 \times 3.14} = 100 

\Rightarrow r = 10 \ cm

(ii) Cone: Radius = 2.5 \ cm , Height = 8 \ cm

\frac{4}{3} \times \pi \times (10)^3 =  n \times \frac{1}{3} \times \pi \times (2.5)^2 \times 8

\Rightarrow n = \frac{4 \times 10^3}{2.5^2 \times 8} = 80

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Question 10: A solid metallic cone, with radius 6 \ cm and height 10 \ cm , is made of some heavy metal A. In order to reduce its weight, a conical hole is made in the cone as, shown and it is completely filled with a lighter metal B. The conical hole has a diameter of 6 \ cm and depth 4 \ cm . calculate the ratio of the volume of metal A to the volume of the metal B in the solid.

Answer:

Cone: Radius = 6 \ cm , Height = 10 \ cm

Conical hole: Radius = 3 \ cm , Height = 4 \ cm

Volume of metal A =  \frac{1}{3} \times \pi \times (6)^2 \times 10 -  \frac{1}{3} \times \pi \times (3)^2 \times 4 = 108 \pi

Volume of metal B =  \frac{1}{3} \times \pi \times (3)^2 \times 4 = \frac{36 \pi}{3}

Therefore Ratio = \frac{108 \pi}{\frac{36 \pi}{3}} = 9:1

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Question 11: A hollow sphere of internal and external 6 \ cm and 8 \ cm respectively is melted and recast into small cones of base radius 2 \ cm and height 8 \ cm . Find the number of cones. [2012]

Answer:

Sphere: Internal radius = 6 \ cm , External radius = 8 \ cm

Cone: Radius = 2 \ cm , Height = 8 \ cm

\frac{4}{3} \times \pi \times (8)^3 - \frac{4}{3} \times \pi \times (6)^3 =  n \times \frac{1}{3} \times \pi \times (2)^2 \times 8

\Rightarrow n = \frac{4 \times (8^3-6^3)}{2^2 \times 8} = 37

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Question 12: The surface area of a solid metallic sphere is 2464\ cm^2 . It is melted and recast into solid right circular cones of radius 3.5 \ cm and height 7 \ cm . Calculate: (i) the radius of the sphere (ii) the number of cones recast. (Take \pi = \frac{22}{7} ) [2014]

Answer:

Surface area of sphere = 2464 \ cm^2

Cone: Radius = 3.5 \ cm , Height = 7 \ cm

(i) 4 \pi r^2 = 2464 \Rightarrow r^2 = \frac{2464 \times 7}{4 \times 22} = 196

Hence R = 14 \ cm

(ii) \frac{4}{3} \times \pi \times (14)^3 = n \times \frac{1}{3} \times \pi \times (3.5)^2 \times 7

\Rightarrow n = \frac{4 \times 14^3}{3.5^2 \times 7} = 128

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