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A balloon is in the form of a right circular cylinder of radius $1.5\,m$ and length $4\,m$ and is surmounted by hemispherical ends. If the radius is increased by $0.01\,m$ and the length by $0.05\,m$, then the percentage change in the volume of the balloon is
(A) $2.38\% $
(B) $2.48\% $
(C) $2.038\% $
(D) $2.58\% $

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Last updated date: 13th Jun 2024
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Answer
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Hint:
In this question first we will find the volume of the right circular cylinder and volume of its hemispherical ends and then we will find the volume of the cylinder and its hemispherical ends when length and radius is changed. And finally we will find the percentage change in the volume of the right circular cylinder.

Formula used:
The volume of the cylinder is given by $\pi {r^2}h$ and the volume of hemisphere is given by $\dfrac{2}{3}\pi {r^3}$ .

Complete step by step solution:
The radius of the cylinder is given $1.5\,m$ and its length is $4\,m$ .
The formula for volume of cylinder is given by $\pi {r^2}h$
Put the value of radius and length in the above formula.
$ \Rightarrow \pi {\left( {1.5} \right)^2}\left( 4 \right) = 9\pi \,{m^3}$
The formula for volume of hemisphere is $\dfrac{2}{3}\pi {r^3}$.
$ \Rightarrow \dfrac{2}{3}\pi {\left( {1.5} \right)^3} = 2.25\pi \,{m^3}$
Therefore, the total volume is $ = 9\pi \,{m^3} + 2.25\pi \,{m^3} = 11.25\pi \,{m^3}$ .
Now, radius becomes $r = 1.5\,m + 0.01\,m = 1.51\,m$ and the length becomes $h = 4\,m + 0.05\,m = 4.05\,m$ . Volume of the cylinder after the increase in radius and length is:
$ \Rightarrow \pi {\left( {1.51} \right)^2}\left( {4.05} \right) = 9.234405\pi \,{m^3}$
Now, the volume of the hemisphere becomes
$ \Rightarrow \dfrac{2}{3}\pi {\left( {1.51} \right)^3} = 2.2953\pi \,{m^3}$
Therefore, the total volume is $ = 9.234405\pi \,{m^3} + 2.2953\pi \,{m^3} = 11.529705\pi \,{m^3}$ .
Now, change in percentage is :
$ \Rightarrow \dfrac{{\left( {11.529705\pi - 11.25\pi } \right)}}{{11.25\pi }} \times 100 = 2.48\% $
Therefore, the change in percentage is $2.48\% $

Hence, the correct option is (B).

Note:
The percentage change in volume is calculated on the original volume of the balloon and it is not calculated on the volume of the balloon after the change in radius and length. The important thing in this question is that we have to find the total volume of the balloon and it depends on the shape of the balloon i.e. in our case the balloon is cylindrical and hemispherical both. So we have to find the volume of cylinder and hemisphere to calculate the total volume of the balloon.