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$\left( a \right)V = \dfrac{{r\left( {A - \pi {r^2}} \right)}}{2}$

$\left( b \right)V = \dfrac{{r\left( {A - 2\pi {r^2}} \right)}}{2}$

$\left( c \right)V = \dfrac{{r\left( {A + 2\pi {r^2}} \right)}}{2}$

$\left( d \right)$ None of these

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Given expressions:

V = $\pi {r^2}h$………………. (1)

And, A = $2\pi {r^2} + 2\pi rh$…………….. (2)

Now we have to find out the value of V in terms of A, $\pi $ and r.

So we have to eliminate the h.

So from equation (2) first find out the value of h we have,

$ \Rightarrow A = 2\pi {r^2} + 2\pi rh$

$ \Rightarrow A - 2\pi {r^2} = 2\pi rh$

Now divide by $2\pi r$throughout we have,

$ \Rightarrow \dfrac{{A - 2\pi {r^2}}}{{2\pi r}} = h$…………………… (3)

Now substitute this value of h from equation (3) in first equation we have,

$ \Rightarrow V = \pi {r^2}\left( {\dfrac{{A - 2\pi {r^2}}}{{2\pi r}}} \right)$

Now simplify the above equation we have,

\[ \Rightarrow V = \dfrac{{r\left( {A - 2\pi {r^2}} \right)}}{2}\]

So this is the required value of V in terms of A, $\pi $ and r.