Question

If the diameter of the cross-section of a wire is decreased by 5% how much percent will the length be increased so that the volume remains the same.

Answer 1

Hint: Here first we will use the formula of volume of the cylinder for standard values of radius and height of the cylinder. Then we will calculate the new radius and substitute it in the formula of volume to get the new height and then calculate the increase in the percentage of height.
The diameter is twice the radius of the cylinder.
\[d = 2r\].

Complete step by step solution:
The volume of the cylinder is given by:
\[V = \pi {r^2}h\]
Where r is the radius and h is the length of the cylinder.
Now since we know that,
\[r = \dfrac{d}{2}\]
Hence, substituting the value of r we get:
\[
  V = \pi {\left( {\dfrac{d}{2}} \right)^2}h \\
  V = \dfrac{1}{4}\pi {d^2}h.............\left( 1 \right) \\
 \]
Now calculating the new diameter we get:
Let new diameter be \[{d^{'}}\]
Since diameter is decreased by 5% therefore,
\[
  {d^{'}} = d - 5\% d \\
  {d^{'}} = d - \dfrac{5}{{100}}d \\
  {d^{'}} = \dfrac{{100d - 5d}}{{100}} \\
  {d^{'}} = \dfrac{{95d}}{{100}} \\
  {d^{'}} = \dfrac{{19d}}{{20}} \\
 \]
Now let the new length be \[{h^{'}}\].
Now calculating new volume of cylinder by substituting the new values of diameter and new height.
Let new volume be \[{V^{'}}\].
Hence,
\[{V^{'}} = \dfrac{1}{4}\pi {\left( {{d^{'}}} \right)^2}{h^{'}}\]
Now substituting the new value of diameter we get:
\[
  {V^{'}} = \dfrac{1}{4}\pi {\left( {\dfrac{{19d}}{{20}}} \right)^2}{h^{'}} \\
  {V^{'}} = \dfrac{1}{4}\pi \left( {\dfrac{{361}}{{400}}} \right){d^2}{h^{'}}.............\left( 2 \right) \\
 \]
Since it is given that old diameter is equal to the new diameter therefore,
Equating equations (1) and (2) we get:
\[
  \left( {\dfrac{{361}}{{400}}} \right){h^{'}} = h \\
  {h^{'}} = \dfrac{{400h}}{{361}} \\
 \]
Now we will calculate the increase in length :
Increase in length = new length – old length
Therefore, increase in length is given by:
\[
  {h^{'}} - h = \dfrac{{400h}}{{361}} - h \\
  {h^{'}} - h = \dfrac{{400h - 361h}}{{361}} \\
  {h^{'}} - h = \dfrac{{39h}}{{361}} \\
 \]
Now we will calculate the percentage increase in length we get:
\[{\text{Percentage increase}} = \dfrac{{{\text{increase in length}}}}{{{\text{old length}}}} \times 100\]
Substituting the values we get:
\[
  {\text{Percentage increase in length}} = \dfrac{{\left( {\dfrac{{39h}}{{361}}} \right)}}{h} \times 100 \\
  {\text{Percentage increase in length}} = \dfrac{{39h}}{{361h}} \times 100 \\
  {\text{Percentage increase in length}} = \dfrac{{39}}{{361}} \times 100 \\
  {\text{Percentage increase in length}} = 10.8\% \\
 \]

Hence the percentage increase in length is \[10.8\% \].

Note:
The volume of cylinder is given by:
\[V = \pi {r^2}h\]
The percentage increase in length is given by:
\[{\text{Percentage increase}} = \dfrac{{{\text{increase in length}}}}{{{\text{old length}}}} \times 100\]