**Problem 1 :**

A hollow cylindrical pipe is of length 40 cm. Its internal and external radii are 4 cm and 12 cm respectively. It is melted and cast into a solid cylinder of length 20 cm. Find the radius of the new solid.

**Solution :**

length of cylindrical pipe (h) = 40 cm

internal radius of the pipe (r) = 4 cm

external radius of the pipe (R) = 12 cm

Height of cylinder = 20 cm

Volume of hollow cylindrical pipe = Volume of cylinder

Π h (R^{2} - r^{2}) = Π r^{2} h

h (R^{2} - r^{2}) = r^{2} h

(40) (12^{2} - 4^{2}) = r^{2} (20)

(40) (144 - 16) = r^{2} (20)

(40) (128)/20 = r^{2}

r^{2 } = (40) (128)/20

r^{2 }= 2 (128)

r = √256

r = 16 cm

Therefore radius of cylinder = 16 cm

**Problem 2 :**

An iron right circular cone of diameter 8 cm and height 12 cm is melted and recast into spherical lead shots each of radius 4 mm. How many lead shots can be made?

**Solution :**

diameter of right circular cone = 8 cm

radius of right circular cone (r) = 4 cm

Height of right circular cone (h) = 12 cm

Radius of spherical lead shot (r) = 4 mm

10 mm = 1 cm

4 mm = (4/10) cm

Volume of right circular cone

= n x (Number of spherical lead shots)

n = Volume of cone/Number of spherical lead shots

n = (1/3)Π r^{2} h/(4/3)Π r^{3}

n = (1/3) (4)^{2} (12) x (3/4) (10/4)^{3}

n = 3 x 5 x 5 x 10

n = 750 lead shots

Therefore 750 lead shots can be made.

**Problem 3 :**

A container with a rectangular base of length 4.4 m and breadth 2 m is used to collect rain water. The height of the water level in the container is 4 cm and water is transferred into a cylindrical vessel with radius 40 cm. What will be the height of the water level in the cylinder?

**Solution :**

Volume of water in cuboidal container = Volume of water in cylindrical container

lwh = Π r^{2}h

l = 4.4 m, w = 2 m, h = 4 cm ==> 4/100 ==> 1/25 m

r = 40 cm ==> 40/100

4.4(2) (1/25) = Π (2/5)^{2}h

h = 4.4 (2) (1/25) (1/Π) (25/4)

h = 2.2/Π

h = 0.7 m

**Problem 4 :**

A cylindrical bucket of height 32 cm and radius 18 cm is filled with sand. The bucket is emptied on the ground and a conical heap and sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.

**Solution :**

Volume of sand in the cylindrical bucket = Volume of sand of conical heap

Let r_{1}, h_{1} be the radius and height of cylindrical tank

Let r_{2}, h_{2} be the radius and height of conical tank

Π r_{1}^{2}h_{1} = (1/3)Π r_{2}^{2}h_{2}

r_{1 }= 18 cm, h_{1 }= 32 cm and h_{2 }= 24 cm

18^{2}(32) = (1/3) r_{2}^{2}(24)

18^{2}(32) = 8 r_{2}^{2}

r_{2}^{2 }= (18⋅18⋅32)/8

r_{2 } = 18(2)

r_{2} = 36 cm

So, the radius of the conical heap is 36 cm.

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