00:01
So in this question, the concentration of a drug in a bloodstream, t hours after injection into a body is given by this function c, and we want to know when is the concentration of the drug in the bloodstream its greatest? we'll round our answer to two decimal places.
00:19
So we're really trying to maximize our concentration, yes? so to do that, we're going to look at our derivative.
00:26
We're going to look at c prime of t.
00:31
To get c prime of t we need the quotient rule the quotient rule says i start with my denominator 0 .4 plus t squared and then i multiply by the derivative of the numerator the derivative of 6t is 6 minus my numerator 6t times the derivative of the denominator that's just 2t all over my denominator my denominator.
01:04
Denominator being squared over 0 .4 plus t squared being squared.
01:13
Now, once i have c prime, what do i have to do? i have to see where c prime is equal to zero.
01:23
So we're going to set this c prime equal to zero.
01:27
Now, how do we set a fraction equal to zero? a fraction is equal to zero only when my numerator is equal to zero.
01:36
So i've got 0 .4 plus t squared times 6 minus the quantity of 6t times 2t equals 0.
01:52
To solve this equation, i'll distribute my 6.
01:55
So 2 .4 plus 6t squared minus 6t times 2t that's minus 12t squared equals 0...