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Question
A pen stand made of wood is in the shape of a cuboid with four conical depressions and a cubical depression to hold the pens and pins, respectively. The dimension of the cuboid are 10 cm, 5 cm and 4 cm. The radius of each of the conical depressions is 0.5 cm and the depth is 2.1 cm. The edge of the cubical depression is 3 cm. Find the volume of the wood in the entire stand.
Solution
Given that, length of cuboid pen stand (l) = 10 cm
Breadth of cubiod pen stand (b) = 5 cm
And height of cuboid pen stand (h) = 4 cm
∴ Volume of cuboid pen stand
= l × b × h
= 10 × 5 × 4
= 200 cm3
Also, radius of conical depression (r) = 0.5 cm
And height (depth) of a conical depression (h1) = 2.1 cm
∴ Volume of a conical depression
= `1/3pir^2h_1`
= `1/3 xx 22/7 xx 0.5 xx 0.5 xx 2.1`
= `(22 xx 5 xx 5)/1000`
= `22/40`
= `11/20`
= 0.55 cm3
Also, given
Edge of cubical depression (a) = 3 cm
∴ Volume of cubical depression = (a)3 = (3) = 27 cm3
So, volume of 4 conical depressions
= 4 × Volume of a conical depression
= `4 xx 11/20`
= `11/5 cm^3`
Hence, the volume of wood in the entire pen stand
= Volume of cuboid pen stand – Volume of 4 conical depressions – Volume of a cubical depressions
= `200 - 11/5 - 27`
= `200 - 146/5`
= 200 – 29.2
= 170.8 cm3
So, the required volume of the wood in the entire stand is 170.8 cm3.
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