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421.
A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see the given Fig.).

Let l, b and h be respectively the length, breadth and height of a cuboid, then
l = 15 cm
b = 10 cm
and h = 3.5 cm
Volume of cuboid
= (l x b x h) cm³ = (15 x 10 x 3.5) cm³= 525 cm³Let r cm be the radius and h cm be the height of conical part, the
r = 0.5 cm, h = 1.4 cm
Now,
Volume = $V o l u m e space space equals space 1 third space πr squared straight h equals space open parentheses 1 third straight x 22 over 7 straight x space 0.5 space straight x space 0.5 space straight x space 1.4 space close parentheses space cm cubed space equals space open parentheses fraction numerator 7.7 over denominator 21 end fraction close parentheses space cm cubed$

Hence, the volume of wood in the entire stand
= Volume of cuboid - 4 (Volume of cone)

$equals space open parentheses 252 space minus space 4 space straight x space 737 over 21 close parentheses space cm cubed equals space left parenthesis 525 space minus 1.466 right parenthesis space cm cubed equals space 523.534 space cm cubed.$

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422.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

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423.

A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8g mass. (Use π = 3.14)

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424.

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

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425. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm³. Check whether she is correct, taking the above as the inside measurements and π = 3.14.

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426.

A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

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427.

Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

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428. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.

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429.

A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

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430.

A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

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