Jay T. answered • 03/09/23
Retired Engineer/Math Tutor
(a)
C(1.5) = 0.047
C(2.0) = 0.060
C(2.5) = 0.071
C(3.0) = 0.079
C(3.5) = 0.083
C(4.0) = 0.085
C(4.5) = 0.083
C(5.0) = 0.040
From the table, t=4 is the time of highest concentration
(b)
By inspection from the graph, t=4 is the time of greatest concentration.
(c)
C(t) = 4t/(125+t3)
C’(t) = 4(125-2t3)/(125+t3)
C’(t) = 0 when 125 = 2t3
t = 3.914
C(3.914) = 0.0855
Since C(3.5) and C(4.0) < C(3.914), we know that t=3.914 is a maximum.
(d)
C'(t) specifies the rate of increase, so C''(t) is used to find the maximum rate.
C’’(t) = -24t2(-t3+250)/(t3+125)3
C’’(t) = 0 when -24t2(-t3+250) = 0
t = 0 or -6.288
Since the measurement interval is [0,5], t=0 is the time when the rate of concentration is greatest. That t=0 is a maximum can be shown by inspection of a graph of C’(t) or computing C’(1) = 0.031 < C’(0) = 0.032.
(e)
This is done in each part.
Jay T.
Hi. In part (c), the critical point was found by taking the 1st derivative, setting it to 0, and solving. Then it was proved that CP was a max. In part (d) the idea was to find where the slope of C'(t) was a max, that being the point of the max rate of increase. Treating C'(t) as a function meant taking it's derivative, finding the max, and proving the CP found was a max. So, C'(t) is used to find the max concentration, C''(t) is used to find the max rate of change of the concentration. Hope this helps.03/09/23
Jay T.
Hi again. I attempted to clarify it better as follows: Part (c): Getting the max concentration means finding the max C(t) in the given interval. That is done by taking the first derivative of C(t), setting it to zero to get the critical point, and solving for t. Then, showing that C(t) at points left and right of the CP is lower proved the CP is a max. Part (d): C'(t) represents the rate of change of concentration with time. Getting the fastest rate of change of rate of change requires taking the derivative of C'(t) which is C''(t), finding the critical point, and verifying it to be a max. Hope all this helps.03/09/23
Baba B.
Could you show the steps of (C) and (D)? I'm at a loss for how you ended with those solutions.03/09/23