Many fundamental laws of physics and chemistry can be formulated when doing this.
To make sure that we have enough resourses to sustain ourselves. You see, if we use up, say, all water on earth, then obviously we won't have any more and we'll die.
The independent variable is how much weight you add to the boat.
The dependent variable is: Did it or did it not sink yet ?
A) 
The total energy of the system is equal to the maximum elastic potential energy, that is achieved when the displacement is equal to the amplitude (x=A):
(1)
where k is the spring constant.
The total energy, which is conserved, at any other point of the motion is the sum of elastic potential energy and kinetic energy:
(2)
where x is the displacement, m the mass, and v the speed.
We want to know the displacement x at which the elastic potential energy is 1/3 of the kinetic energy:

Using (2) we can rewrite this as

And using (1), we find

Substituting
into the last equation, we find the value of x:

B) 
In this case, the kinetic energy is 1/10 of the total energy:

Since we have

we can write

And so we find:
