I assume the equation is $\displaystyle x = At^be^{-ct}$. Using the derivative, express the maximum point (which is t = 30) through b and c. This gives you one equation. For the second equation, you can take x(4t) = x(t) / 2 where t = 30. After some cancellations, this gives you another linear equation on b and c. From these two equations you can find b and c. Then find A from x(30) = 0.2.
The answer is below (for checking only).
Spoiler:
If you can't use derivatives, you can assume that the maximum of $\displaystyle At^be^{-ct}$ occurs at t = b / c. Draw several graphs for various values of b and c and verify that this is the case. The rest of the recommendations in post #2 should still work.