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(B) The amplitude of a damped oscillator is given by the equation $A = A_0 e^{-\lambda t}$,where $A_0$ is the initial amplitude,$\lambda$ is the dampi

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$2^3$

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(B) The amplitude of a damped oscillator is given by the equation $A = A_0 e^{-\lambda t}$,where $A_0$ is the initial amplitude,$\lambda$ is the damping constant,and $t$ is time.Given that at $t = 1 	ext{ min}$,the amplitude becomes half:$\frac{A_0}{2} = A_0 e^{-\lambda(1)}$$\frac{1}{2} = e^{-\lambda} \Rightarrow e^{\lambda} = 2$.Now,for $t = 3 	ext{ min}$,the amplitude $A$ is:$A = A_0 e^{-\lambda(3)} = A_0 (e^{-\lambda})^3 = A_0 \left(\frac{1}{e^{\lambda}}ight)^3$Substituting $e^{\lambda} = 2$:$A = A_0 \left(\frac{1}{2}ight)^3 = \frac{A_0}{2^3}$.Comparing this to $\frac{A_0}{X}$,we get $X = 2^3$.

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

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