The number of applications for patents, N, grew dramatically in recent years, wi

Question

The number of applications for patents, N, grew dramatically in recent years, with growth averaging about 3.5% per year. That is, N'(t) = 0.035N(t). a) Find the function that satisfies this equation. Assume that t = 0 corresponds to 1980, when approximately 111,000 patent applications were received. b) Estimate the number of patent applications in 2020. c) Estimate the doubling time for N(t). a) Choose the correct answer below. A. N(t) = 111,000 e^0.035t B. N'(t) = 111,000 e^0.035t C. N(t) = 111,000t^0.035 e D. N(t) = 0.035 e^111,000t

Explanation / Answer

A).

Suppose 0.035 = k, then

N'(t) = k*N(t)

d(N(t))/dt = k*N(t)

d(N(t))/N(t) = k*dt

Using integration both side

ln (N(t)) = kt + C

N(t) = e^(kt + C)

N(t) = e^C*e^(kt)

given that at t = 0, N = 111000 patent

111000 = e^C*e^(k*0)

e^C = 111000, So

N(t) = 111000*e^(kt)

since k = 0.035, So

N(t) = 111000*e^(0.035t)

The correct option is A.

B).

in 2020, t = 2020 - 1980 = 40

N(40) = 111000*e^(0.035*40)

N(40) = 450127.19

C).

when N(t) is double, then N(0)

2*N(0) = N(0)*e^(0.035*t)

(remember that N(0) = 111000)

e^(0.035*t) = 2

t = (1/0.035)*ln 2

t = 19.80 years

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