If an investment of d dollars is compounded continuously at an annual interest r

Question

If an investment of d dollars is compounded continuously at an annual interest rate r, then the amount of money A(t) earned at time t > 0 (in years) is the solution of the initial value problem A'(t) = rA(t) .A(0) = d where r should be expressed as a fraction rather than as a percentage rate, e.g., an annual interest rate of r = 12% would be expressed as r 12/100 Suppose you have just placed d dollars in a bank account that pays 5 percent (annual) interest, compounded continuously. How much do you have in the account after 4 years? AMOUNT = dollars (Do not include dollar signs in your answer.) How long (in years) will it take your money to double? TIME = years How long (in years) will it take your money to triple? TIME = years

Explanation / Answer

A' = r A(t) dA/dt = r A(t) so ln A(t) / A(0) = r t so A(t) = A(0) e ^ r t a) t= 4 A(0) =d r = 0.05 so A(t) = d e^ 0.2 b) t = ln 2 / 0.05 t= 13.86 years c) t = ln 3/0/.05 t =21.97 years

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