Optical Physics; Light Entering an Atmosphere

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[SOLVED] Optical Physics; Light Entering an Atmosphere

Homework Statement
Quoted word for word from the question sheet:

A light ray enters the atmosphere of a planet and descends vertically 20.0 km to the surface.
The index of refraction where the light enters the atmosphere is 1.000, and it increases linearly towards the surface where it has a value of 1.005.
a) How long does it take the ray to traverse this path?
b) Compare this to the time it takes in the absence of an atmosphere.

The attempt at a solution
Right, I know (or at least, I'm fairly sure that I do) the final answer, just not how to properly obtain it.
I know that the velocity of light through a medium is related to the refractive index by v=c/n, and I know that the time it takes for the light to travel a certain distance is given by t=s/v.

By working out the change in n per metre, and plugging it into a program I whipped up in MatLAB, I got a time of 6.683x10^-5s, which seems reasonable compared to the answer of 6.67*10^-5 I got for part b)

So I'm just stumped on how to get the answer without doing it the long way. Something tells me to use integration, but I've tried that with t=(s*n)/c and I end up with an answer 200 times too small.

Many thanks in advance for any help rendered, I'm sure that once it's explained I'll be kicking myself :P
 
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Let n=1+.005t/T.
20=\int_0^T(c/n)dt.
Solve for T.
 
Many thanks! Got the answer now :)

I knew that integrating was involved, I just couldn't make that last leap :D
 

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