Question

When the circumference of a toy balloon is increased from 20 cm to 25 cm, then its radius (in cm) is increased by
(a) 5
(b) \[\dfrac{5}{\pi }\]
(c) \[\dfrac{5}{2\pi }\]
(d) \[\dfrac{\pi }{5}\]

Answer 1

Hint: Let us take the rogh figures of two balloons as follows

We solve this problem by assuming that there are two balloons of circumferences 20 cm and 25 cm. then we use the formula of the circumference of two balloons.
The formula of the circumference of a sphere having the radius \[r\]is given as
\[C=2\pi r\]
Then by finding the radii of two spheres we can calculate the difference in radii.
Complete step by step answer:
We are given that the circumference of the balloon is increased from 20 cm to 25 cm.
Let us assume that the first balloon having a circumference of 20 cm.
Let us assume that the radius of initial balloon as \[r\]
We know that the formula of circumference of sphere having the radius \[r\]is given as
\[C=2\pi r\]
By using this formula we get
\[\begin{align}
  & \Rightarrow 20=2\pi r \\
 & \Rightarrow r=\dfrac{20}{2\pi } \\
\end{align}\]
Now, let us assume that the circumference of final balloon as 25 cm.
Let us assume that the radius of the final balloon as \[R\]
Now, by using the circumference formula we get
\[\begin{align}
  & \Rightarrow 25=2\pi R \\
 & \Rightarrow R=\dfrac{25}{2\pi } \\
\end{align}\]
We are asked to find the increase in the radius.
Now, by subtracting the initial radius from the final radius we get
\[\begin{align}
  & \Rightarrow R-r=\dfrac{25}{2\pi }-\dfrac{20}{2\pi } \\
 & \Rightarrow R-r=\dfrac{5}{2\pi } \\
\end{align}\]
Therefore we can conclude that the increase of the radius in the balloon is \[\dfrac{5}{2\pi }\]
So, option (c) is correct answer.

Note:
Students may do mistakes in taking the formula of the circumference of the sphere.
The formula of the circumference of a sphere having the radius \[r\]is given as
\[C=2\pi r\]
Here, we can see that it is the same as the circumference of the circle.
But students may do mistake and take the formula as
\[C=4\pi r\]
This is the wrong answer because both the circle and sphere have the same formula for circumference. There is no change in the circumference. But there is a change in the area.
The area of the circle is given as
\[A=\pi {{r}^{2}}\]
The area formula of the sphere is given as
\[A=4\pi {{r}^{2}}\]