Question

A $1$ cubic centimetre of copper is drowned into a wire of $0.2\,mm$ diameter. Find the length of the wire.

Answer 1

Hint: In the given question, we are provided with the volume of copper used in making a wire. So, we have to find the length of the wire given the diameter of the wire. So, we must remember the formula for volume of a cylinder to solve the problem. We will substitute the value of known quantities in the formula to find the unknown quantity and get the required result.

Complete step by step answer:
A wire is cylindrical in shape. A cylinder is a three dimensional solid geometrical figure with straight parallel sides and a circular base. In the problem, we are given the volume of copper used in making the wire. We know that the volume of the wire is the same as the volume of material used in making it. So, the volume of wire is $1\,c{m^3}$. Now, we are also given the diameter of the wire as $0.2mm$. We know that radius is half of the diameter of a shape.
So, we get the radius of the wire $ = \dfrac{{0.2}}{2}\,mm = 0.1\,mm$.

Now, we have to calculate the length of the copper wire formed. We know that the volume of a cylinder is given by $\pi {r^2}h$, where r is the radius of the cylinder and h is the height of a cylinder. Hence, Volume of wire $ = \pi {r^2}h$
Putting in the value of known quantities radius and volume, we get,
$ \Rightarrow \pi {\left( {0.1\,mm} \right)^2} \times h = 1\,c{m^3}$
Now, we know that one centimetre equals $10$ millimetres. So, $1c{m^3} = 1000\,m{m^3}$.
Hence, we get,
$ \Rightarrow \pi {\left( {\dfrac{1}{{10}}mm} \right)^2} \times h = 1000\,m{m^3}$

Substituting the value of $\pi $ as $\left( {\dfrac{{22}}{7}} \right)$, we get,
\[ \Rightarrow \dfrac{{22}}{7} \times \dfrac{1}{{100}} \times h = 1000\]
Shifting all the constants to right side of the equation, we get,
\[ \Rightarrow h = 1000 \times 100 \times \dfrac{7}{{22}}\,mm\]
Simplifying the expression, we get,
\[ \Rightarrow h = 31818.18\,mm\]
Now, we convert the length of wire from millimetres to metres. We know that $1$ metre consists of $1000$ millimetres. So, we have,
\[ \therefore h = 31.81818\,m\]

So, the length of the wire is approximately $31.82$ metres approximately.

Note: We must know the formulae for area, volume and perimeter of basic shapes like square, rectangle, parallelogram, circle, etc. One must have a strong grip over concepts of transposition in order to solve equations formed in the problem. We should substitute the value of $\pi $ correctly in the equation to get to the right answer. We also must take care of the calculations while doing such questions.