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To see this in pretty print, you can visit this link:
i=integrate (E^(-.05x)) dx from (.5x+7) to (2*x-14)Evaluating that for any age would give the area under the curve corresponding to that age.
The next step, then, would be to find the maximum area for any age.
To do this, we should be able to take the derivative of that previous equation, and set it equal to 0 in order to maximize it. The upper age limit is 2t-14 (as you note earlier in your comment) rather than the 2t 14 you mention later.
Even though I am too lazy to do the graphing, I am inclined, based on intuition, to believe the latter curves rather than the former.Any other comments and corrections are very welcome!