Question

The radius of a wire is decreased to one-third. If volume remains the same the length will become
A. \[{\text{1 time}}\]
B. \[{\text{3 times}}\]
C. \[{\text{6 times}}\]
D. \[{\text{9 times}}\]

Answer 1

Hint: Assume \[r\] and \[h\] be the radius and length of a wire and we know that wire is of cylinder shape .Use the formula of volume of wire , which is volume of cylinder of radius \[r\] and height \[h\] , \[V = \pi {r^2}h\] to solve the question ,then we compare initial and final heights(here length of wire) to get the answer.

Complete step by step answer:
According to the question, the radius of a piece of wire is decreased to one third 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 \[V = \pi {r^2}h\].Now as per the question, radius is reduced to one third, let the new radius be \[r' = \dfrac{r}{3}\] and the new height be \[h'\]. Refer to the figure shown below for better understanding-

Therefore, the new volume is \[V = \pi r{'^2}h'\]
Now since the volume remains constant ,
\[ \Rightarrow \pi {r^2}h = \pi r{'^2}h' - (1)\]
Now, put \[r' = \dfrac{r}{3}\] in equation (1), we get-
\[ \Rightarrow \pi {r^2}h = \pi {\left( {\dfrac{r}{3}} \right)^2}h'\]
\[ \Rightarrow h = \dfrac{1}{9}h'\]
\[ \therefore h' = 9h\]
Hence, the new height is 9 times the original height.This means the length of the wire became 9 times the original length with the same volume.

Therefore, the correct option is D.

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 = $r/3$. Solve it to find the value of the new height.