00:01
So for this problem, we know that for an element to decay proportionally to its mass, we're going to be able to write a function relating our mass to our rate of decay, as the rate of change of our mass with respect to time is equal to a sum negative constant k times our mass m.
00:19
And so from here, we know that we're going to be able to integrate this using a separation of variables.
00:24
So essentially we're just going to divide both sides by m and multiply by d t.
00:28
So we get 1 over m dm is equal to negative k d t.
00:33
And from here we can go ahead and integrate.
00:36
And we know the integral of 1 over m is just the natural log of m.
00:41
And this is going to be, oops, let me rewrite that m.
00:44
And this is going to be equal to negative kt plus our constant of integration c.
00:51
And so from here we're going to be able to rewrite this as m is equal to e to the negative kt plus c power.
01:00
Or addressed m is equal to a .e to the negative kt power.
01:05
And in order to solve for our constants or our coefficients, a and k, we're going to be able to use the information that we're given...