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Find the number of coins, 2.5 cm in diameter and 0.4 cm thick, to be melted to form a right circular cylinder of height 15 cm and diameter 8 cm.
(a) 324
(b) 372
(c) 368
(d) 384
(e) none of these.

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Last updated date: 20th Jun 2024
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Answer
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Hint: To solve this problem we will first find the volume of the one coin. As coins are in cylindrical shape so their volume we will calculate using the formula $\pi {{r}^{2}}h$. After finding the volume of one coin we will assume the number of coins required to be n, and multiply the number n with the volume of one coin. After doing this we will equate the obtained expression with the volume of the right circular cylinder which we will find using the same formula i.e. $\pi {{r}^{2}}h$. From there we will solve to find the value of n, and that will be our required answer.

Complete step by step answer:
We are given a coin whose diameter is 2.5 cm and 0.4 cm,
And we have to find the number of such coins required to form a right circular cylinder of height 15 cm and diameter 8 cm.
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So first of all we will find the radius of the coin as,
Radius of coin = $\dfrac{Diameter\,of\,coin}{2}=\dfrac{2.5}{2}=1.25\,cm$
Now we know that coins are of cylindrical shape so their volume will be given by the formula $\pi {{r}^{2}}h$ where r is radius and h is height of the coin,
So we get the volume of the coin as,
= $\pi {{r}^{2}}h$
= $\pi \times {{\left( 1.25 \right)}^{2}}\times 0.4$
= $\pi {{\left( 1.25 \right)}^{2}}\left( 0.4 \right)$
Now we will assume that number of coins required to form the cylinder to be n,
So if we multiply the volume of one coin with n then we will get the volume of n coins,
Hence we get,
Volume of n number of coins = $n\pi {{\left( 1.25 \right)}^{2}}\left( 0.4 \right)$
Now this obtained volume should be equal to the volume of the right circular cylinder,
So we have to now find the volume of the right circular cylinder using the formula $\pi {{r}^{2}}h$,
So first radius of the cylinder is given as = $\dfrac{Diameter\,of\,cyl\operatorname{in}der}{2}=\dfrac{8}{2}=4\,cm$
Hence volume of cylinder we get as,
= $\pi {{r}^{2}}h$
= $\pi {{\left( 4 \right)}^{2}}\left( 15 \right)$
Now equation the volume of the cylinder with the volume of n number of coins, we get
\[\begin{align}
  & n\pi {{\left( 1.4 \right)}^{2}}\left( 0.4 \right)=\pi {{\left( 4 \right)}^{2}}\left( 15 \right) \\
 & \Rightarrow n=\dfrac{\pi {{\left( 4 \right)}^{2}}\left( 15 \right)}{\pi {{\left( 1.4 \right)}^{2}}\left( 0.4 \right)} \\
 & \Rightarrow n=\dfrac{{{\left( 4 \right)}^{2}}\left( 15 \right)}{{{\left( 1.4 \right)}^{2}}\left( 0.4 \right)} \\
 & \Rightarrow n=384 \\
\end{align}\]
Hence we get the number of coins as 384.

So, the correct answer is “Option B”.

Note: To solve this problem you should have prior knowledge of how to calculate the volume of the cylindrical figure. And also before solving these kinds of problems you need to first analyse the question carefully that which parameter here has to be considered like in this question it was volume, some students may get confused and might take surface area instead of volume but it is wrong so read the problem carefully first.