Answer:
It will take 14 years before the investment triples
Step-by-step explanation:
Continuous Compounding
Is the mathematical limit that compound interest can reach if it was calculated and reinvested into an account's balance over a theoretically infinite number of periods.
The formula for continuous compounding is derived from the formula for the future value of a compound interest investment:
[tex]FV = PV\cdot e^{i.t}[/tex]
Where:
FV = Future value of the investment
PV = Present value of the investment
i = Interest rate
t = Time
It's required to find the time for an investment to triple, that is, FV = 3 PV, knowing the interest rate is i=8%=0.08.
Substituting the known values:
[tex]3PV = PV\cdot e^{i.t}[/tex]
Dividing by PV:
[tex]3 = e^{i.t}[/tex]
Taking logarithms:
[tex]\ln 3=i.t[/tex]
Solving for t:
[tex]\displaystyle t=\frac{\ln 3}{i}[/tex]
[tex]\displaystyle t=\frac{\ln 3}{0.08}[/tex]
t = 13.7 years
Rounding up:
It will take 14 years before the investment triples
It will take 37.5 years before the investment triples.
The formula for calculating the future value is expressed as:
[tex]A =Pe^{rt}[/tex]
P is the principal
r is the rate (in %)
t is the time (in years)
If the investment tripled, then A = 3P. The equation will become:
[tex]3P=Pe^{rt}\\3=e^{rt}\\ln3 =lne^{rt}\\rt = 3\\0.08t =3\\t =\frac{3}{0.08} \\t =37.5 years[/tex]
Hence it will take 37.5 years before the investment triples.
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