Question

The radius of a piece of wire is decreased to one half. If volume remains the same, then its length will increase
A) 2 times
B) 3 times
C) 4 times
D) 5 times

Answer 1

Hint – Assume r and h be the radius and length of a wire. Use the formula of volume of wire $ = \pi {r^2}h$ to solve the question.

Complete step-by-step answer:
According to the question, the radius of a piece of wire is decreased to one half and the volume remains the same.
So, let us assume the radius of the wire be r and the length of the wire be h.
We know, volume of the wire $ = \pi {r^2}h$.
Now, as per the question, radius is reduced to one half.
So, let the new radius be $r' = \dfrac{r}{2}$ and the new height be h’.
Refer to the figure shown below for better understanding-

Therefore, the new volume is $ = \pi r{'^2}h'$
Now, since the volume remains constant.
$ \Rightarrow \pi {r^2}h = \pi r{'^2}h' - (1)$
Now, put $r' = \dfrac{r}{2}$ in equation (1), we get-
$
   \Rightarrow \pi {r^2}h = \pi {\left( {\dfrac{r}{2}} \right)^2}h' \\
   \Rightarrow h = \left( {\dfrac{1}{4}} \right)h' \\
   \Rightarrow h' = 4h \\
$
Hence, the new height is 4 times the original height.
So, the correct option is (C) 4 times.

Note – Whenever solving such types of questions, write down the conditions and the values given in the question. Now, we know from the question that the volume remains constant, so keep the new volume equal to the original volume and substitute the new radius = r/2. Solve it to find the value of new height.