Page 172 - How To Solve Word Problems In Calculus

ln 2 = 0.05t

ln 2
t = ≈ 13.86 years
0.05

EXAMPLE 7
How much money should Ariel invest in a bank account pay-
ing 8 percent annual interest compounded continuously if she
wants to use the money to buy a $20,000 car in 4 years?

Solution
A
rt − rt
The equation A = Pe is equivalent to P = = Ae .
e rt
P = Ae − rt

= 20,000e −(0.08)(4)
= 20,000e −0.32

= 20,000(0.726149)
= $14,522.98

Additional Exponential Models
Exponential functions are prevalent in many problems in
science, business, economics, medicine, and sociology. One
important function, the logistic function, is often used to ana-
lyze population growth that is limited by natural environ-
mental factors. The logistic function is also used to analyze
the spread of epidemics and the propagation of rumors.
The general form of the logistic function is
B
P(t) = where A, B, and k are usually determined
1 + Ae − Bkt
experimentally. Once these values are found, the logistic func-
tion can answer many questions concerning growth.

EXAMPLE 8
Environmentalists predict that in t years the deer population
20
in a forest will be P(t) = thousand.
1 + 4e −2t
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