Think of the Sun low in the sky, shortly before sunset…

Think of the Sun low in the sky, shortly before sunset. Its light travels through a very long stretch of air before it reaches your eyes. On the way the air takes light out of the beam by scattering. For each colour, the fraction of the light that is left at the end of the stretch is $e^{-\tau}$, where $\tau$ is in proportion to $1/\lambda^4$. A measurement shows that red light of wavelength $700$ nm has $80\ \%$ of its intensity left at the end of the stretch. The student wants to understand why the Sun looks red then, but she has only the one measurement, for red light. Use this to predict what happens to the other colours. Find the number of times that violet light of $400$ nm is scattered more strongly than the red. Then find the percentage of violet light, and of blue light of $450$ nm, that is left.