A young person with no initial capital invests k dollars per year in a retiremen

Question

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.

a) Write a differential equation which models the rate of the change of the sum S(t) with t in years (this will involve the parameter k). S' = ?

b) Use part a) to determine a formula for the sum S(t) -- (this will involve the parameter k): S(t) = ?

c) What value of k will provide 2261000 dollars in 45 years? k =?

Explanation / Answer

If S(t) is the amount of money accumulated in time t years then,

then dS/dt = k + 0.06*S ................ ANS (a)

here dS/dt is in dollars per year and k dollars will be incremented with 0.06*S every year.

This can be rewritten as -

dS = k +0.06*S dt

or dS / (k + 0.06*S) = dt

Now integrating both sides, we have

ln(k + 0.06*S) / 0.06 = t + C

or ln(k + 0.06*S) = 0.06t + C

or k + 0.06*S = e^(0.06t + C)

or k + 0.06*S = C*e^(0.06t)

or S = ( Ce^(0.06t) - k ) /0.06    ............(1)

Now to find value of C, let t = 0 and S = 0, we get

0 = ( C - k)/0.06 => C = k

By substituting the value of C in (1), we get

S(t) = (ke^(0.06t) - k)/0.06

       =k(e^(0.06t) -1)/0.06 ............. ANS (b)

or k = 0.06*S/( e^(0.06t) -1)

Now we can find k by substituting 2261000 for S and 45 for t.

k = 135660 / e^1.7
   = 135660 / 5.473947

or k = 24782.85 dollars
  

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