Solução_Calculo_Stewart_6e

F.
TX.10
SECTION 3.8 EXPONENTIAL GROWTH AND DECAY ¤ 117
results in 0.05(10W ) − 0.001(10W )W =0 ⇔ 0.5W − 0.01W 2 =0 ⇔ 50W − W 2 =0 ⇔
W (50 − W )=0 ⇔ W =0or 50. SinceC =10W , C =0or 500. Thus, the population pairs (C, W ) that lead to
stable populations are (0, 0) and (500, 50).Soitispossibleforthetwospeciestoliveinharmony.
3.8 Exponential Growth and Decay
1. The relative growth rate is 1 P
dP
dt
Thus, P (6) = 2e 0.7944(6) ≈ 234.99 or about 235 members.
=0.7944,so
dP
dt =0.7944P and, by Theorem 2, P (t) =P (0)e0.7944t =2e 0.7944t .
3. (a) By Theorem 2, P (t) =P (0)e kt = 100e kt . NowP (1) = 100e k(1) = 420 ⇒ e k = 420
100
⇒ k =ln4.2.
So P (t) =100e (ln 4.2)t =100(4.2) t .
(b) P (3) = 100(4.2) 3 =7408.8 ≈ 7409 bacteria
(c) dP/dt = kP ⇒ P 0 (3) = k · P (3) = (ln 4.2) 100(4.2) 3 [from part (a)] ≈ 10,632 bacteria/hour
(d) P (t) = 100(4.2) t =10,000 ⇒ (4.2) t =100 ⇒ t =(ln100)/(ln 4.2) ≈ 3.2 hours
5. (a) Let the population (in millions) in the year t be P (t). Since the initial time is the year 1750, we substitute t − 1750 for t in
Theorem 2, so the exponential model gives P (t) =P (1750)e k(t−1750) .ThenP (1800) = 980 = 790e k(1800−1750)
⇒
980
= 790 ek(50) ⇒ ln 980 =50k ⇒ k = 1 980
ln ≈ 0.0043104. Sowiththismodel,wehave
790 50 790
P (1900) = 790e k(1900−1750) ≈ 1508 million, and P (1950) = 790e k(1950−1750) ≈ 1871 million. Both of these
estimates are much too low.
(b) In this case, the exponential model gives P (t) =P (1850)e k(t−1850) ⇒ P (1900) = 1650 = 1260e k(1900−1850) ⇒
ln 1650 = k(50) ⇒ k = 1 1650
ln ≈ 0.005393. Sowiththismodel,weestimate
1260 50 1260
P (1950) = 1260e k(1950−1850) ≈ 2161 million. This is still too low, but closer than the estimate of P (1950) in part (a).
(c) The exponential model gives P (t) =P (1900)e k(t−1900) ⇒ P (1950) = 2560 = 1650e k(1950−1900) ⇒
ln 2560 = k(50) ⇒ k = 1 2560
1650 50
ln
1650
≈ 0.008785. With this model, we estimate
P (2000) = 1650e k(2000−1900) ≈ 3972 million. This is much too low. The discrepancy is explained by the fact that the
world birth rate (average yearly number of births per person) is about the same as always, whereas the mortality rate
(especially the infant mortality rate) is much lower, owing mostly to advances in medical science and to the wars in the first
part of the twentieth century. The exponential model assumes, among other things, that the birth and mortality rates will
remain constant.
7. (a) If y =[N 2O 5] then by Theorem 2, dy
dt = −0.0005y ⇒ y(t) =y(0)e−0.0005t = Ce −0.0005t .
(b) y(t) =Ce −0.0005t =0.9C ⇒ e −0.0005t =0.9 ⇒ −0.0005t =ln0.9 ⇒ t = −2000 ln 0.9 ≈ 211 s