Question

How many silver coins, 1.75 cm in diameter and of thickness 2mm, must be melted to form a cuboid of dimensions \[5.5cm\times 10cm\times 3.5cm\]?

Answer 1

Hint: First of all, take the number of silver coins as ‘n’. Now, find the volume of ‘n’ coins and cuboid and equate them, that is, Volume of Cuboid = n (Volume of each coin). Here, find the value of n which is the number of silver coins.

Complete step-by-step answer:
Here, we are given that some silver coins of diameter 1.75 cm and thickness 2 mm are melted to form a cuboid of dimensions \[5.5cm\times 10cm\times 3.5cm\]. Here, we need to find the number of silver coins. Before proceeding with the question, we must know that when one object of a particular shape is converted into another object/objects of the same or different shape, then the volume always remains constant. That is the volume of the object before and after the conversion would be the same while the other quantities such as surface area, length, breadth, radius, etc. could change.

Now, let us consider our question. Let us assume that there are a total of ‘n’ silver coins. Now, these ‘n’ silver coins of diameter 1.75 cm and thickness 2 mm are melted to form a cuboid of dimensions \[5.5cm\times 10cm\times 3.5cm\].
First of all, let us find the total volume of ‘n’ coins. We know that the coins are cylindrical and the volume of the cylinder is given by \[\pi {{r}^{2}}h\].
By substituting the value of \[r=\dfrac{1.75}{2}=0.875cm\] and \[h=\dfrac{2}{10}cm=0.2cm\]. We get, Volume of one coin,
\[{{V}_{s}}=\pi {{\left( 0.875 \right)}^{2}}.\left( 0.2 \right)=0.153125\pi \text{ }c{{m}^{3}}\]
So, the volume of ‘n’ coins would be, \[=n{{V}_{s}}=n\left( 0.153125\pi \text{ }c{{m}^{3}} \right).....\left( i \right)\]
Now, let us find the volume of the cuboid. We know that the volume of the cuboid is given by \[L\times B\times H\].
By substituting the values of L = 5.5 cm, B = 10 cm and H = 3.5 cm, we get,
The volume of Cuboid, \[{{V}_{c}}=5.5\times 10\times 3.5=192.5\text{ }c{{m}^{3}}.....\left( ii \right)\]
As we already know that when an object is melted into another object, then volume remains constant. So, we get,
(Volume of ‘n’ coins) = (Volume of the cuboid)
\[n{{V}_{s}}={{V}_{c}}\]
By substituting the respective values from equation (i) and equation (ii), we get,
\[n.\left( 0.153125 \right).\pi =192.5\]
By substituting \[\pi =\dfrac{22}{7}\], we get,
\[n\left( 0.153125 \right)\times \dfrac{22}{7}=192.5\]
\[n\left( 0.48125 \right)=192.5\]
By dividing both the sides by 0.48125, we get,
\[n=\dfrac{192.5}{0.48125}=400\]
Hence, we get the number of silver coins as 400.

Note: Students often make this mistake of using the formulas of volume, area, etc. without properly looking whether the radius or diameter is given in the question. They must properly read and if the diameter is given, first convert it into radius and then only put it into the formula to get the desired value. Students must remember that whenever any 3-dimensional object is converted into another, then the volume always remains constant. Also, always remember to first convert all the parameters like radius, height, etc. in the same unit and then only solve the problem.