Page 173 - How To Solve Word Problems In Calculus

(a) What is the current population?
(b) What will be the population at the end of each year for the
first three years?
(c) What is the population growth rate after three years?
(d) When does the population growth rate begin to decline?
(e) What is the population limit of the deer in this forest?
(f ) When will the deer population reach 80 percent of its
limit?
Solution
P(t)
The graph of the deer 20
population is shown to 17.5
15
the right.
12.5
(a) P(0) = 4 thousand deer 10
7.5
(b) P(1) = 12.98 thousand deer 5
P(2) = 18.63 thousand deer 2.5
P(3) = 19.80 thousand deer 1 2 3 4 5 t
(c) The rate of population

growth is given by r(t) = P (t).
20
)
P(t) = = 20(1 + 4e −2t −1
1 + 4e −2t

P (t) =−20(1 + 4e −2t −2 (−8e −2t )
)
160e −2t
=
)
(1 + 4e −2t 2
After three years the population rate of change is
160e −6

P (3) = ≈ 0.389. The rate of growth is 389 deer
)
(1 + 4e −6 2
per year.
(d) The rate of population growth increases when r (t) > 0 and

decreases when r (t) < 0. The population growth rate be-

gins to decline when r (t) = P (t) = 0.
d d
)
)
(1 + 4e −2t 2 (160e −2t ) − (160e −2t ) (1 + 4e −2t 2
dt dt
P (t) =
)
(1 + 4e −2t 4
) (−320e
(1 + 4e −2t 2 −2t )−(160e −2t )(2)(1+4e −2t )(−8e −2t )
=
(1 + 4e −2t 4
)
160