Unformatted text preview: nuously, what
must be your initial principal to meet your 10-year objective?
b) Referring to18a), if the best investment available to you will only pay 5% annual interest,
show that your initial principal must be increased by more than 10% over the answer to a)
in order to reach the $10,000 objective in 10 years.
19. a) If you deposit $5000 in an account earning 5% annual interest compounded continuously,
how long will it take for the principal to reach $7500?
b) If the deposit of $5000 earns 10% interest, how long will it take for the principal to reach
28 2 Differential Equations
20. A 5-year investment of $10,000 begins at 6% annual interest compounded continuously. The
investment agreement calls for the interest rate to be evaluated after 2½ years, when it can
move up or down by at most 2% for the remaining 2½ years. Determine the minimum and
maximum principal you can receive at the end of 5 years.
21. For each of the interest rates listed in Table 2.1, find the number of years for an initial principal
22. a) Suppose you deposit an initial principal of $1000 in an account paying 6% annual interest
compounded continuously. Every year, including the first, you make additional deposits of
$500 spread out uniformly throughout the year. By repeating the arguments that led to
(2.8), show that the principal P(t ) satisfies the ODE
= .06 P + 500 .
dt (2.11) b) Find the solution of (2.11) satisfying the initial condition P(0) = 1000 .
c) After 10 years what will be the total principal in the account as described in a)? How much
of this total has come from deposits? How much from interest?
23. a) Consider the investment scenario described in22a), except that your initial principal is zero.
Each year you want to make a deposit of K dollars spread out uniformly throughout the
year, so that at the end of 25 years the account will have a principal of $50,000. By
repeating the arguments that led to equation (2.8), show that the principal P(t ) satisfies the
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