If a spherical balloon has a variable diameter $3 x + \frac{9}{2}$ , then the rate of change of its volume with respect to x is

27 π (2 x + 3)^{2}

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\frac{27 π}{16} (2 x + 3)^{2}

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\frac{27 π}{8} (2 x + 3)^{2}

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Solution

The correct option is C $\frac{27 π}{8} (2 x + 3)^{2}$
Radius of balloon $= r = \frac{3}{4} (2 x + 3) \Rightarrow \frac{d r}{d x} = \frac{3}{2}$
$∴$ Rate of change in volume = $4 π {(\frac{3}{4})}^{2} (2 x + 3)^{2} . \frac{3}{2}$
$= \frac{27 π}{8} (2 x + 3)^{2}$

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