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(B) In damped oscillation,the amplitude $a$ at time $t$ is given by the formula:$a = a_0 e^{-bt}$where $a_0$ is the initial amplitude and $b$ is the d

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$8$

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(B) In damped oscillation,the amplitude $a$ at time $t$ is given by the formula:$a = a_0 e^{-bt}$where $a_0$ is the initial amplitude and $b$ is the damping constant.Given that the amplitude becomes half in $1$ minute ($t = 1$ min):$\frac{a_0}{2} = a_0 e^{-b(1)}$$e^{-b} = \frac{1}{2}$We need to find the amplitude after $3$ minutes ($t = 3$ min),which is given as $\frac{a_0}{x}$:$\frac{a_0}{x} = a_0 e^{-b(3)}$$\frac{1}{x} = (e^{-b})^3$Substituting the value of $e^{-b} = \frac{1}{2}$:$\frac{1}{x} = (\frac{1}{2})^3 = \frac{1}{8}$Therefore,$x = 8$.

The amplitude of a damped oscillator becomes half in $1$ minute. The amplitude after $3$ minutes will be $\frac{1}{x}$ times the original. Then $x$ is

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