The amplitude of a damped oscillator becomes half in one minute. The amplitude after 3 minutes will be 1 x…

In damped oscillation, amplitude changes according to following formula-

$a=a_0{e}^{-bt}$

Here $a_0=$Initial amplitude

$a=$Amplitude at any time $t$

Using given data-

$\frac{a_0}{2}=a_0{e}^{-b\times 1} \dots \left(1\right)$

$\mathrm{and} \frac{a_0}{x}=a_0{e}^{-b\times 3} \dots \left(2\right)$

From equation $\left(1\right)-$

$\Rightarrow e^{-b}=\frac{1}{2}$

Using equation $\left(1\right) \mathrm{and} \left(2\right)-$

$\Rightarrow \frac{x}{2}=\frac{e^{-b}}{e^{-3b}}=\frac{e^{-b}}{{\left(e^{-b}\right)}^3}$

$\Rightarrow \frac{x}{2}=\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{{\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^3}=4$

$\Rightarrow x=8 \min$