Question

A copper rod of diameter $1cm$ and length $8cm$ is drawn into a wire of length $18m$ of uniform thickness. Find the thickness of wire.

Answer 1

Hint: Here, converting the copper rod into wire does not affect its volume. So, equating the volumes of the copper rod and drawn wire we can get the solution very easily. You can also eliminate the constant values as they anyways get cancelled on simplification.

Complete step-by-step solution:
A copper rod of diameter $1$cm and length $8$cm is drawn into a wire of length $18$m of uniform thickness. When the copper rod is melted or drawn into a long wire, the copper content is neither added or removed. It means that the volume of the copper will still remain constant. Now we are going to use this logic to solve this problem.
Before that, let us note down all the given values in a systematic order as follows:
Length of the copper rod $(l)$ $ = 8cm$
Diameter of the copper rod $ = 1cm$
Radius of the copper rod$(r)$$ = 0.5cm$
Length of the wire $(l) = 1800cm$ $(1m = 100cm)$
Now, we need to assume the radius of the wire to be some variable.
Let radius of the wire be $R$
After everything we set down, now we need to equate the values of volumes of both the structures as we discussed previously.
Then,
$
  \pi {r^2}l = \pi {R^2}L \\
\Rightarrow {(0.5)^2} \times 8 = {R^2} \times 1800 \\
   \Rightarrow {R^2} = 2 \div 1800 \\
\Rightarrow {R^2} = \dfrac{1}{{900}} \\
\Rightarrow R = \sqrt {\dfrac{1}{{900}}} \\
\Rightarrow R = \dfrac{1}{{30}}
 $
So we will get the value of $R = 0.033cm$
Therefore,
We got the value of the radius of the wire. But in the question, we are asked to find out the thickness of the wire. And it is not that hard to get that as all of us know that,
Thickness is the diameter of the wire.
So,
Thickness $ = 2 \times R$
$ = 2 \times 0.033$
$ = 0.066cm$

So, the required thickness of the wire is $0.066cm$.

Note: If we have the same wire and then it is given the shape of the circle and the rectangle, then we must know that the perimeter of the rectangle and the circumference of the circle will be equal as the wire length is the same in both the cases. Similarly if we have the material and convert it into a different shape then the volume will be the same.