Question

# Given: V = $\pi {r^2}h$ and A = $2\pi {r^2} + 2\pi rh$ find V in terms of A, $\pi$ and r.$\left( a \right)V = \dfrac{{r\left( {A - \pi {r^2}} \right)}}{2}$ $\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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Hint: In this particular question use the concept that we have to eliminate h so from the given expressions find out the value of h from any expression and substitute in the other expression and simplify, so use these concepts to reach the solution of the question.

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.
Hence option (a) is the correct answer.

Note: Whenever we face such types of questions the key concept involved in this question is elimination a particular variable using substitution method so first find out the value of h from equation (2) as above and substitute in equation (1) and simplify we will get the required value of V in terms of A, $\pi$ and r.