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How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions $5.5\ cm \times 10\ cm \times 3.5\ cm$?
Given:
Diameter of each coin $=1.75\ cm$
The thickness of each coin $=2\ mm$
Dimensions of the cuboid are \( 5.5 \mathrm{~cm} \times 10 \mathrm{~cm} \times 3.5 \mathrm{~cm} \).
To do:
We have to find the number of coins that must be melted to form the cuboid.
Solution:
Radius of the coin $r=\frac{1.75}{2}$
$=\frac{175}{100 \times 2}$
$=\frac{7}{8} \mathrm{~cm}$
The thickness of the coin $h=2 \mathrm{~mm}$
$=\frac{2}{10} \mathrm{~cm}$
$=\frac{1}{5} \mathrm{~cm}$
The volume of each coin $=\pi r^{2} h$
$=\frac{22}{7} \times (\frac{7}{8})^{2} \times \frac{1}{5}$
$=\frac{22}{7} \times \frac{7}{8} \times \frac{7}{8} \times \frac{1}{5}$
$=\frac{77}{160} \mathrm{~cm}^{3}$
Volume of the cuboid $=5.5 \times 10 \times 3.5$
$=192.5 \mathrm{~cm}^{3}$
Therefore,
Number of coins formed $=$ Volume of the cuboid $\div$ Volume of each coin
$=\frac{\frac{192.5}{77}}{160}$
$=\frac{192.5 \times 160}{77}$
$=2.5 \times 160$
$=400$
The number of coins that must be melted to form the cuboid is $400$.