When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account, so that if the amount present is P, then dP/dt = kP (k constant) where P is in dollars, t is in years, and k is a constant.
(a) Solve this differential equation to find the relationship.
(b) Use properties of logarithms and exponential functions to write P as a function of t.
(c) If $50,000 is invested (when t = 0) and the amount in the account after 10 years is $135,914, find the function that gives the value of the investment as a function of t.
(d) In part (c), what does the value of k represent?