(a)
(b)
(a)
(b)
Then
P(t) = (3.5 × 10^9 )e 0.019t.The question is: for what t does P(t) equal 5.5 × 10^15 ? We solve
Taking the logarithm of each side yields
where it seems reasonable to round off as we have. Thus, if the human
population continued to grow at the present rate, there would be one person for
every square foot of land in the year 2720.
Case II: Restricted Growth
The rate of change of a quantity y = f (t) may be proportional, not to the amount
present, but to a difference between that amount and a fixed constant. Two
situations are to be distinguished: The rate of change is proportional to
a fixed constant A minus the amount of the quantity present:
f ′(t) = k [A − f (t)]
the amount of the quantity present minus a fixed constant A:
f ′(t) = −k [f (t) − A]where (in both) f (t) is the amount at time t and k and A are both positive. We
may conclude that
f(t) is increasing (Fig. N9–8a):
f (t) = A − ce−kt
f(t) is decreasing (Fig. N9–8b):for some positive constant c.
f (t) = A + ce−kt