Value of \(t = 21.98\)years when an investment in a savings account grows to three times the initial value after t years with rate of interest is 5%, compounded continuously.
What is compounded continuously?
" Compounded continuously is defined as when on certain amount process of calculating interest by reinvesting the amount for infinite number of years."
Formula used
\(A = Pe^{rt}\)
\(A =\) Final amount
\(P=\)initial amount
\(r=\) rate of interest
\(t=\)time period
According to the question,
\('P'\) represents the initial value in the saving account
\('A'\) represents the final amount
As per the question
\(A = 3P\)
Rate of interest \('r' = 5\%\)
Time \(= t\) years
Substitute the value in the formula of compounded continuously we get,
\(3P = P e^{\frac{5}{100}( t)} \\\\\implies 3 = e^{0.05t} \\\\\implies log 3 = (0.05t)(loge)\\\\\implies 0.05t =\frac{ log3 }{loge} \\\\\implies 0.05t = \frac{0.4771}{0.434} \\\\\implies t = \frac{1.0993}{0.05}\\ \\\implies t = 21.98 years\)
Hence, value of \(t = 21.98\)years when an investment in a savings account grows to three times the initial value after t years with rate of interest is 5%, compounded continuously.
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