00:01
So here we're told that sugar transforms into dextrose at a rate proportional to the amount of sugar.
00:06
So that means the derivative of the amount of sugar with respect to time is equal to a constant k times the current amount of sugar, s.
00:15
So we can isolate s and t in this equation.
00:19
We divide by s, multiply by dt, to get d .s over s equals kdt.
00:24
And that allows us to integrate both sides.
00:27
The integral of the left -hand side is log of s, and the integral of the right -hand side is kt plus a constant.
00:34
And theoretically, we could add a constant on the left, too, but when we move it to the right -hand side, it just becomes a new constant.
00:40
Now we exponentiate both sides to give us s equals e to the k -t plus c, and the right -hand side there is equal to e -to -k -t times e -to -c, and we can call e to the c some new constant a.
00:54
So s is just equal to a, e to the kt.
00:59
And then we're told that the amount of sugar, when t is zero, is 75 grams.
01:04
When we plug in zero to our equation, that just leaves us with a.
01:08
So that means 75 equals a.
01:11
And then we're told s of 30, the amount of sugar, after 30 minutes, well, eight grams have been transformed...