hw3 Problem 2 Previous Problem List Next 1 point A young per

hw3: Problem 2 Previous Problem List Next (1 point) A young person with no initial capital invests k dollars per year in a retirement account at an annual rate of return 0.06. Assume that investments are made continuously and that the return is compounded continuously. Determine a formula for the sum SCt) (this will involve the parameter k): What value of k will provide 4326000 dollars in 47 years?

Solution

from given question the differntial equation is written as: dS/dt = 0.06S + k

k is a constant amount invested every year : dS /(0.06S + k) = dt

integrate both sides: 1/(0.06S + k) dS = 1dt
(1/0.06) ln(0.03S + k) = t + C
ln(0.06S + k) = 0.06t + 0.06C
0.06S + k = e^(0.06t) e^(0.06C)

initial conditions: S(0) = 0

0.06(0) + k = e^(0) e^(0.06C)
e^(0.06C) = k
0.06S + k = ke^(0.06t)
S = k(e^0.06t - 1)/0.06
S(t) = 100k(e^0.06t - 1)/6
S(t) = (1/6)k(100e^0.06t - 100)

and k - ke^(0.06t) = -0.06S
k(1 - e^0.06t) = -0.06S
k = 0.06S / [e^(0.06t) - 1]
k = 0.06(4326000) / [e^(0.06x 47) - 1]
k = $16451.95