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 the wire.

Answer 1

Hint:- Find the volume of rod and wire.
As we given that,
Diameter of the rod is \[1cm\].

So, the radius of the rod will be \[\dfrac{1}{2}cm\].
Length of rod is \[8cm\].
Length of wire is \[18m = 18 \times 100cm = 1800cm\].
So, the thickness of the wire will be \[t{\text{ }}cm\].
As we know that the thickness of wire is equal to its diameter.
So, the diameter of the wire will be \[t{\text{ }}cm\].
And the radius of the wire will be \[\dfrac{t}{2}cm\].
As we know, the shape of rod and wire is always a cylinder.
According to the formula of volume of cylinder.
Volume of the cylinder is \[\pi {r^2}h\], where r is the radius and h will be the height of the cylinder.
So, volume of copper rod will be \[\pi {\left( {\dfrac{1}{2}} \right)^2}8c{m^2} = 2\pi c{m^2}\].
And, volume of wire will be \[\pi {\left( {\dfrac{t}{2}} \right)^2}1800{\text{ }}c{m^2} = 450\pi {t^2}c{m^2}\].
As, we know that if a wire is drawn into a rod.
Then the volume of wire will be equal to the volume of the rod.
So, \[2\pi c{m^2} = 450\pi {t^2}c{m^2}\] (1)
On solving the above equation. We get,
\[ \Rightarrow {t^2} = \dfrac{{2\pi }}{{450\pi }} = \dfrac{1}{{225}}\]
Taking square both sides of the above equation. We get,
\[ \Rightarrow t = \dfrac{1}{{15}} = 0.0666\]\[\]
Hence, the thickness of the given wire will be \[0.0666cm\].
Note:- Whenever we came up with this type of problem then first, we find the
volume of rod and wire using formula to find the volume of cylinder and then,
After comparing the volume of rod and wire we will get the thickness of wire.
Because their volume must be equal because wire is drawn inside the rod. And this will be the
easiest and efficient way to find the thickness of wire.