Damped SHM

The amplitude of damped oscillation is given by,
$A = A_{0}e^{-k t}$
Where $k$ is the damping constant and $k_{medium}$ $\alpha$ $\eta_{medium}$
In air,
$8 = 10e^{-k_{air}(40)}$
$\therefore k_{air} = \dfrac{\ln(5) - 2\ln(2)}{40}$

In $CO_{2}$ ,
$5 = 10e^{-k_{CO_{2}}t}$
$t = \dfrac{\ln(2)}{k_{CO_{2}}}$
$\dfrac{k_{air}}{k_{CO_{2}}} = \dfrac{\eta{air}}{\eta{CO_{2}}} = 1.3$
$\therefore t = 1.3 \times \dfrac{\ln(2)}{k_{air}}$
Substituting value of $k_{air}$
$t = 40 \times 1.3 \times \dfrac{\ln(2)}{\ln(5)-2\ln(2)} = \dfrac{52\ln(2)}{\ln(5)-2\ln(2)}$
$\therefore t \approx 161.59$
Rounding to the greatest integer less than or equal to it,
$t \approx 161$