Question

A continuous annuity is a steady stream of money that is paid out, for example, by making an initial deposit in an account and then making steady withdrawals to pay the annuity. Suppose that an initial deposit of $3600 is made into an account that earns 6% interest compounded continuously, and immediately continuous withdrawals are begun at the rate of $200 per year.  

a.) Set up the differential equation that is satisfied by the amount f(t) of money in the account at time t.

b.) Determine f(t).

c.) With the help of a graphing tool, plot f(t) and approximate the time it will take the account to reach $10,000.

Answer 1

since we should take change in Money = interest savings = 0.0by + 3600 separation of variobles, we get dy sdt - 0.oby +3600 Let u= 0.064 +36002 - du= 0.06 ay Ju Oy 0.06 Placing @ values in dy t 0 = dt 0.06 +3600 u adt 0.06 Now integrating Both sides. Sigusot du ce x 0.06

=) ence)=t+a 0.06 - 16.6 en/o.oby + 36001 = ttc 1 16.6 en 10.06y + 36001 = t + c loobyt 36001-cx e 166 ooby = cх 16.6 2600 y = cх e 7/16.6 - cxe - 3600 0.06 y = cte - 60,000 Now we are given with initial condition 2007 - 4 (0) = cх e - 60000= 200. c=200 + 60000 C = 60200 +/ 16.6 4= 60200 e - 60,000 y = f(t) t/16.6 f(t) = 60200 e - 60,000