Very well done! I started with the equation
, then substituted in the equation for d, brought the
over to the left and integrated. Optical depth would then be equal to the negative of the resulting
on the left hand side. I then solved for opacity.
On the second part, I did a similar procedure except I solved for
using the found value for opacity. You instead equivalently used the mean density over the column and multiplied it by the change in height. Very good! Unfortunately, I believe you miscalculated on your very last line the final answer for the second part.
Alright, your turn!