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.).

Solution

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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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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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.).

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