Question

How many coins 2mm thick and 1.5cm in diameter should be melted in order to form a right circular cylinder having base diameter 6cm and height 8cm?
\[\begin{align}
  & A.\text{ }640 \\
 & B.\text{ }540 \\
 & C.\text{ }740 \\
 & D.\text{ }840 \\
\end{align}\]

Answer 1

Hint: In this question, we are given dimensions of coin and dimensions of a right circular cylinder into which coins will melt to form the shape. Since, some number of coins are melted to form a right circular cylinder, therefore volume of coins will be equal to volume of the formed right circular cylinder. To find the number of coins, we will divide the volume of the right circular cylinder by volume one coin. Since, a coin is in the form of a right circular cylinder therefore, its volume will be calculated using the formula for finding the volume of the right circular cylinder. Volume of the right circular cylinder is given by $V=\pi {{r}^{2}}h$ where, V is the volume, r is the radius of the base of the cylinder and h is the height of the cylinder.

Complete step by step answer:
Dimensions of coin are as follows:
Diameter of the coin is equal to 1.5cm and the height of the coin is equal to 2mm.
Since, units of diameter and height are different, so let us change units of height from millimeter to centimeter.
As we know, 1cm = 10 mm, therefore,
$2mm=\dfrac{2}{10}cm=0.2cm$.
Therefore, diameter = 1.5cm and height = 0.2cm.
We require radius for calculating volume, so let us find radius by dividing diameter by 2. We get radius as $\dfrac{1.5}{2}cm$. Hence, $\dfrac{1.5}{2}cm$ and h = 0.2cm.
Since, coin is in the shape of right circular cylinder and volume of right circular cylinder is given as $\pi {{r}^{2}}h$ so, volume of coin will be equal to $\pi {{r}^{2}}h$.
Putting values of r and h, we get:
\[\begin{align}
  & \text{Volume of coin}=\pi {{r}^{2}}h \\
 & \Rightarrow \pi \times \left( \dfrac{1.5}{2} \right)\left( \dfrac{1.5}{2} \right)\left( 0.2 \right) \\
 & \Rightarrow \pi \times \dfrac{0.45}{4} \\
 & \Rightarrow 0.1125\pi c{{m}^{3}} \\
\end{align}\]
Dimensions of formed right circular cylinder are as follows:
Diameter of the right circular cylinder is equal to 6cm and the height of the right circular cylinder is equal to 8cm.
Volume of the right circular cylinder is $\pi {{r}^{2}}h$ so we need radius. Thus, let us find radius by dividing diameter by 2, we get radius as \[\dfrac{6}{2}=3cm\].
Therefore, r = 3cm and h = 8cm.
\[\begin{align}
  & \text{Volume of right circular cylinder}=\pi {{r}^{2}}h \\
 & \Rightarrow \pi {{\left( 3 \right)}^{2}}\times 8 \\
 & \Rightarrow \pi \times 9\times 8 \\
 & \Rightarrow 72\pi c{{m}^{3}} \\
\end{align}\]
Let n be the number of coins which are melted to form the right circular cylinder, therefore, the volume of n coins will be equal to the volume of the right circular cylinder.
\[\text{Volume of one coin}\times n=\text{Volume of right circular cylinder}\]. Therefore,
\[\begin{align}
  & n=\dfrac{\text{Volume of right circular cylinder}}{\text{Volume of one coin}} \\
 & \Rightarrow n=\dfrac{72\pi c{{m}^{3}}}{0.1125\pi c{{m}^{3}}} \\
 & \Rightarrow n=\dfrac{720000}{1125} \\
 & \Rightarrow n=640 \\
\end{align}\]

As n was supposed to be the number of coins. Therefore, 640 coins should be melted to form the required right circular cylinder.

Note: Students should note that units of all dimensions should be the same. While dividing also, units of volume of the right circular cylinder should be equal to the volume of the coin. Don't forget to find the radius of the base of the cylinder. Students should keep in mind all the formulas for finding volume of three dimensional shapes. We have not substituted the value of $\pi $ because we know that it will be cancelled later and hence, to ease our calculations we took $\pi $ as it is.