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 \mathrm{~cm} \times 5 \mathrm{~cm} \times 4 \mathrm{~cm} $. The radius of each of the conical depression is $ 0.5 \mathrm{~cm} $ and the depth is $ 2.1 \mathrm{~cm} $. The edge of the cubical depression is $ 3 \mathrm{~cm} $. Find the volume of the wood in the entire stand.
Given:
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 \( 10 \mathrm{~cm} \times 5 \mathrm{~cm} \times 4 \mathrm{~cm} \).
The radius of each of the conical depression is \( 0.5 \mathrm{~cm} \) and the depth is \( 2.1 \mathrm{~cm} \).
The edge of the cubical depression is \( 3 \mathrm{~cm} \).
To do:
We have to find the volume of the wood in the entire stand.
Solution:
Length of the cuboid pen stand $l = 10\ cm$
Breadth of the cuboid pen stand $b = 5\ cm$
Height of the cuboid pen stand $h = 4\ cm$
Therefore,
Volume of the cuboid pend stand $= lbh$
$= 10 \times 5 \times 4$
$= 200\ cm^3$
Radius of the conical depression $r = 0.5\ cm$
Height of the conical depression $h_1 = 2.1\ cm$
Volume of the conical depression $=\frac{1}{3} \pi r^2 h_1$
$=\frac{1}{3} \times \frac{22}{7} \times (0.5)^2 \times 2.1$
$=\frac{22 \times 5 \times 5}{1000}$
$=\frac{22}{40}$
$=\frac{11}{20}$
$=0.55 \mathrm{~cm}^{3}$
Edge of the cubical depression $a=3 \mathrm{~cm}$
Volume of the cubical depression $=(a)^{3}$
$=(3)^{3}$
$=27 \mathrm{~cm}^{3}$
Volume of four conical depressions $=4 \times$ Volume of the conical depression
$=4 \times \frac{11}{20}$
$=\frac{11}{5} \mathrm{~cm}^{3}$
Volume of wood in the entire pen stand $=$ Volume of the cuboidal pen stand $-$ Volume of 4 conical depressions $-$ Volume of the cubical depression
$=200-\frac{11}{5}-27$
$=200-\frac{146}{5}$
$=200-29.2$
$=170.8 \mathrm{~cm}^{3}$
The volume of the wood in the entire stand is \( 170.8 \mathrm{~cm}^{3} \).
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