Thank you so much for your solution, Jeremy ! Really appreciate your time and help. :)

Just wondering if there's any way to solve this without using calculus ?

I have given my reply in the second answer. See if you can understand. The key is to know that there is a 'double root equation' -(m-10t)^2.

Double root equation does not cross the x axis (parameter t in this case) graphically. This means if we solely plot -(m-10t)^2..without the k, the top of the curve (or the maximum) touches P=0 at t=3. We make use of this fact (P=0 at t=3, for p=-(m-10t)^2 without the k) to find m.

From m, you can find k.