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]

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]

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.

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 $108 \ cm$.

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.

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?

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.    [2010]

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.    [2011]

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]

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.

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]

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]

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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