If an investment of d dollars is compounded continuously at an annual interest r
ID: 2890295 • Letter: I
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>0 (in years) is the solution of the initial value problem A'(t) = rA(t) A(0) d 12 where r should be expressed as a action rather than as a percentage rate, eg an annual interest rate of r-12% would be expressed as F-100 Suppose you have just placed d dolars in a bank account that pays 5 percent (annual) terest, compounded continuously (a) How much do you have in the account after 8 years? AMOUNT Do not include doliar signs in your answer) ll dollans b) How long (in years) will it take your money to double? years (c) How long (in years) will take your money to triple? TIME Ill years Important: On an exam you may be expected to know the equations shown in the initial value problem above. They may not be supplied to youExplanation / Answer
dA/dt = rA
So, dA/A = rdt
Integrating :
ln|A|= rt + C
A = e^(rt + C)
A = Ce^(rt)
Let initial amt be A0, then
A = A0 * e^(rt)
So, we have
A = d*e^(0.05*8)
A = de^(0.4) ----> ANS
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Doubling time :
2d = de^(0.05t)
2 = e^(0.05t)
ln(2) = 0.05t
t = ln(2) / 0.05
i.e approx
13.8629 years ----> ANS
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Triple :
ln(3) / 0.05
21.9722 yrs ----> ANS