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How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form cuboid of dimensions
Answer
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Hint: Use the fact that the total volume of all coins melted will be equal to the volume of cuboid being made. Each coin can be considered as a cylinder and its volume can be calculated using $V=\pi {{r}^{2}}h$. The volume of the cuboid can be calculated using the formula $V=l\times b\times h$.
“Complete step-by-step answer:”
A coin can be considered to be a cylinder with the height of the cylinder as the thickness of the coin and the diameter as given.
Hence, the volume of one coin can be calculated using the formula for volume of a cylinder, $V=\pi {{r}^{2}}h$.
The radius $r=\dfrac{d}{2}$ where d is the given diameter
Hence, \[r=\dfrac{1.75cm}{2}\]
\[\begin{align}
& \Rightarrow r=\dfrac{17.5mm}{2} \\
& \Rightarrow r=8.75mm \\
\end{align}\]
Using this radius and height $h=2mm$in the above formula, we get
$\begin{align}
& V=\pi {{r}^{2}}h \\
& \Rightarrow V=\pi {{\left( 8.75mm \right)}^{2}}\left( 2mm \right) \\
& \Rightarrow V=\pi \left( 76.5625m{{m}^{2}} \right)\left( 2mm \right) \\
& \Rightarrow V=\dfrac{22}{7}\left( 153.125m{{m}^{3}} \right) \\
& \Rightarrow V=481.25m{{m}^{3}} \\
\end{align}$
Hence the volume of one coin is $481.25m{{m}^{3}}$.
Let us now assume that the number of coins required to be melted is n.
Thus, the volume of n coins will be equal to the volume of the cuboid.
The volume of cuboid is given by the formula, $V=l\times b\times h$
Putting the values of $l=5.5cm,\ b=10cm,\ h=3.5cm$ in the above formula we get
$\begin{align}
& V=5.5cm\times 10cm\times 3.5cm \\
& \Rightarrow V=192.5c{{m}^{3}} \\
\end{align}$
This volume is now equal to the volume of n coins. Using the volume of coin and cuboid we get,
$n\left( 481.25m{{m}^{3}} \right)=192.5c{{m}^{3}}$
Converting $c{{m}^{3}}$ into $m{{m}^{3}}$,
\[\begin{align}
& n\left( 481.25m{{m}^{3}} \right)=192.5\times 1000m{{m}^{3}} \\
& \Rightarrow n=\dfrac{192500m{{m}^{3}}}{481.25m{{m}^{3}}} \\
& \Rightarrow n=400 \\
\end{align}\]
Thus n = 400 coins are required to be melted to form a cuboid of the given dimensions.
Note: It is important to keep the units in mind while doing this question. While calculating the volume of the coin, the radius is in cm while the thickness is in mm. Hence, all dimensions should be carefully converted in the same unit before calculating the values. Similarly, the conversions need to be carefully carried out at other places as well, as required.
“Complete step-by-step answer:”
A coin can be considered to be a cylinder with the height of the cylinder as the thickness of the coin and the diameter as given.
Hence, the volume of one coin can be calculated using the formula for volume of a cylinder, $V=\pi {{r}^{2}}h$.
The radius $r=\dfrac{d}{2}$ where d is the given diameter
Hence, \[r=\dfrac{1.75cm}{2}\]
\[\begin{align}
& \Rightarrow r=\dfrac{17.5mm}{2} \\
& \Rightarrow r=8.75mm \\
\end{align}\]
Using this radius and height $h=2mm$in the above formula, we get
$\begin{align}
& V=\pi {{r}^{2}}h \\
& \Rightarrow V=\pi {{\left( 8.75mm \right)}^{2}}\left( 2mm \right) \\
& \Rightarrow V=\pi \left( 76.5625m{{m}^{2}} \right)\left( 2mm \right) \\
& \Rightarrow V=\dfrac{22}{7}\left( 153.125m{{m}^{3}} \right) \\
& \Rightarrow V=481.25m{{m}^{3}} \\
\end{align}$
Hence the volume of one coin is $481.25m{{m}^{3}}$.
Let us now assume that the number of coins required to be melted is n.
Thus, the volume of n coins will be equal to the volume of the cuboid.
The volume of cuboid is given by the formula, $V=l\times b\times h$
Putting the values of $l=5.5cm,\ b=10cm,\ h=3.5cm$ in the above formula we get
$\begin{align}
& V=5.5cm\times 10cm\times 3.5cm \\
& \Rightarrow V=192.5c{{m}^{3}} \\
\end{align}$
This volume is now equal to the volume of n coins. Using the volume of coin and cuboid we get,
$n\left( 481.25m{{m}^{3}} \right)=192.5c{{m}^{3}}$
Converting $c{{m}^{3}}$ into $m{{m}^{3}}$,
\[\begin{align}
& n\left( 481.25m{{m}^{3}} \right)=192.5\times 1000m{{m}^{3}} \\
& \Rightarrow n=\dfrac{192500m{{m}^{3}}}{481.25m{{m}^{3}}} \\
& \Rightarrow n=400 \\
\end{align}\]
Thus n = 400 coins are required to be melted to form a cuboid of the given dimensions.
Note: It is important to keep the units in mind while doing this question. While calculating the volume of the coin, the radius is in cm while the thickness is in mm. Hence, all dimensions should be carefully converted in the same unit before calculating the values. Similarly, the conversions need to be carefully carried out at other places as well, as required.
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