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(D) Let the volume of the soap bubble be $V$ and the rate of increase be constant, so $V = kt$, where $k$ is a constant.Since $V = \frac{4}{3}\pi R^3$

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None of these

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(D) Let the volume of the soap bubble be $V$ and the rate of increase be constant, so $V = kt$, where $k$ is a constant.Since $V = \frac{4}{3}\pi R^3$, we have $\frac{4}{3}\pi R^3 = kt$, which implies $R = \left( \frac{3k}{4\pi} t \right)^{1/3}$.The pressure inside a soap bubble is given by $P_{in} = P_{atm} + \frac{4T}{R}$, where $P_{atm}$ is atmospheric pressure and $T$ is surface tension.Substituting $R$ in the pressure equation: $P_{in} = P_{atm} + \frac{4T}{\left( \frac{3k}{4\pi} t \right)^{1/3}}$.This can be written as $P_{in} = P_{atm} + C \cdot t^{-1/3}$, where $C$ is a constant.None of the provided graphs ($P$ vs $1/t$, $P$ vs $\log(t)$, or $P$ vs $t$) represent the relationship $P \propto t^{-1/3}$.Therefore, the correct option is $D$.

$A$ soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume with time at a constant rate. The graph that correctly depicts the time dependence of pressure inside the bubble is given by

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