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

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

Sphere: Radius $\displaystyle = 15 \text{ cm }$

Cone: Radius $\displaystyle = 2.5 \text{ cm }$ and Height $\displaystyle = 8 \text{ cm }$

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

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

Internal diameter $\displaystyle = 4 \text{ cm } \Rightarrow$ Internal radius $\displaystyle = 2 \text{ cm }$

External diameter $\displaystyle = 8 \text{ cm } \Rightarrow$ External radius $\displaystyle = 4 \text{ cm }$

Radius of the $\displaystyle \text{ cone } = 4 \text{ cm }$

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

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

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

Internal radius $\displaystyle = 3 \text{ cm }$

External radius $\displaystyle = 5 \text{ cm }$

Height of the cone $\displaystyle = 32 \text{ cm }$

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

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

Therefore $\displaystyle r = 3.5 \text{ cm }$ and hence diameter $\displaystyle = 7 \text{ cm }$

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

Let the radius of the cone $\displaystyle = r$

Height of the cone $\displaystyle = 108 \text{ cm }$

Bigger cone: Height $\displaystyle = 9 \text{ cm }$ and radius $\displaystyle = 20 \text{ cm }$

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

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

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

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

Dimension of the block $\displaystyle = 49 cm \times 44 cm \times 18 \text{ cm }$

Let the radius of the sphere $\displaystyle = r \text{ cm }$

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

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

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

Bowl: Internal radius $\displaystyle = 9 \text{ cm }$

Cone: radius $\displaystyle = 1.5 \text{ cm }$ , Height $\displaystyle = 4 \text{ cm }$

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

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

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Question 7: A hemispherical bowl of diameter $\displaystyle 7.2 \text{ cm }$ is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius $\displaystyle 4.8 \text{ cm } .$ Find the height of the cone if it is completely filled. [2010]

Hemisphere: Radius $\displaystyle = 3.6 \text{ cm }$

Cone: Radius $\displaystyle = 4.8 \text{ cm }$ , Height $\displaystyle = h$

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

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

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Question 8: A solid cone of radius $\displaystyle 5 \text{ cm }$ and height $\displaystyle 8 \text{ cm }$ is melted and made into small spheres of radius $\displaystyle 0.5 \text{ cm } .$ Find the number of spheres formed. [2011]

Cone: Radius $\displaystyle = 5 \text{ cm }$ , Height $\displaystyle = 8 \text{ cm }$

Sphere: Radius $\displaystyle = 0.5 \text{ cm }$

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

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

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

Surface area $\displaystyle = 1256 \text{ cm}^2$

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

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

$\displaystyle \Rightarrow r = 10 \text{ cm }$

(ii) Cone: Radius $\displaystyle = 2.5 \text{ cm }$ , Height $\displaystyle = 8 \text{ cm }$

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

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

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Question 10: A solid metallic cone, with radius $\displaystyle 6 \text{ cm }$ and height $\displaystyle 10 \text{ cm }$ , 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 $\displaystyle 6 \text{ cm }$ and depth $\displaystyle 4 \text{ cm } .$ calculate the ratio of the volume of metal A to the volume of the metal B in the solid.

Cone: Radius $\displaystyle = 6 \text{ cm }$ , Height $\displaystyle = 10 \text{ cm }$

Conical hole: Radius $\displaystyle = 3 \text{ cm }$ , Height $\displaystyle = 4 \text{ cm }$

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

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

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

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

Sphere: Internal radius $\displaystyle = 6 \text{ cm }$ , External radius $\displaystyle = 8 \text{ cm }$

Cone: Radius $\displaystyle = 2 \text{ cm }$ , Height $\displaystyle = 8 \text{ cm }$

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

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

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

Surface area of sphere $\displaystyle = 2464 \text{ cm}^2$
Cone: Radius $\displaystyle = 3.5 \text{ cm }$ , Height $\displaystyle = 7 \text{ cm }$
$\displaystyle \text{(i) } 4 \pi r^2 = 2464 \Rightarrow r^2 = \frac{2464 \times 7}{4 \times 22} = 196$
Hence $\displaystyle R = 14 \text{ cm }$
$\displaystyle \text{(ii) } \frac{4}{3} \times \pi \times (14)^3 = n \times \frac{1}{3} \times \pi \times (3.5)^2 \times 7$
$\displaystyle \Rightarrow n = \frac{4 \times 14^3}{3.5^2 \times 7} = 128$