Question

The diameter of a copper sphere is \[{\mathbf{18cm}}\].The sphere is melted and is drawn into a long wire of uniform circular cross-section. If the length of the wire is \[{\mathbf{108m}}\], find its diameter.

Answer 1

Hint: In this problem we first convert the unit of the diameter of the copper sphere from ’cm’ to ‘m’. Then getting the half of diameter of the copper sphere which is the radius of the copper sphere. Then we calculate the volume of copper sphere by applying the formula of sphere and also calculate the volume of wire by applying the formula of volume of cylinder (In this problem we take wire as a cylindrical shape). After that we equate the volume of copper sphere and volume of the wire of uniform cross section. After calculation we get the radius of the wire. Then we get the required diameter of the wire as it is double of the radius of the wire.

Complete step-by-step answer:
In the given problem,
The diameter of the copper sphere is \[18cm\].
 $ = 18cm $
 $ = \dfrac{{18}}{{100}}m = 0.18m $
Radius of the copper sphere (R) $ = \dfrac{{0.18}}{2} = 0.09 $ m
Length of wire (h) $ = 108m $
Now we applying the formula of volume of sphere
Volume of copper sphere $ = \dfrac{4}{3}\pi {R^3} $
Putting the value of r in the above formula
 \[ = \dfrac{4}{3}\pi {\left( {0.09} \right)^2}\]
   $ = \dfrac{4}{3}\pi \times 0.09 \times 0.09 $
On dividing 0.09 by 3 we get
  $ = 4\pi \times 0.09 \times 0.03 $
On multiplying the numerical terms
Therefore, the volume of copper sphere $ = 0.0108\pi $ $ {m^3} $
Now we applying the formula of volume of cylinder
Volume of wire $ = \pi {r^2}h $
Putting the value of h in the above formula
 $ = \pi {r^2} \times 108 $
 $ = 108\pi {r^2} $ $ {m^3} $
Now we equate the volume of copper with the volume of wire because the copper sphere is melted and drawn into a wire therefore their volume should be equal.
Volume of wire = volume of copper sphere
 $ 108\pi {r^2} $ = $ 0.0108\pi $
Now for calculate the value of ‘r’, $ 108\pi $ is taking in denominator of $ 0.0108\pi $
 $ \Rightarrow {r^2} = \dfrac{{0.0108\pi }}{{108\pi }} $
 $ \Rightarrow {r^2} = 0.0001 $
By taking square root on both sides
\[ \Rightarrow r = \sqrt {0.0001} \]
\[ \Rightarrow r = 0.01\] $ m $
 $ \therefore $ The radius of wire is $ .01m $
Then the diameter of wire $ = 2r $
On putting the value of ‘r’
Diameter of wire $ = 2 \times 0.01 $ $ = 0.02m $
Hence, the diameter of a copper sphere is \[18cm\]. The sphere is melted and is drawn into a long wire of uniform circular cross-section. If the length of the wire is \[108m\], the diameter of wire is $ 0.02m $ .
So, the correct answer is “$ 0.02m $”.

Note: In this given problem is based conversion of the shape of the one solid into another solid by melting. In this process the volume of the both solids should be equal. To solve this problem we should know the formula of volume of solids.