In a bank, a principal P (in Rs.) increases continuously at the rate of 8% per year, that is, \(\displaystyle\frac{dP}{dt} = \frac{8}{100}P\), where t is the time in years. The amount to which Rs. 500 grows in 25 years is (take \(e^2 = 7.389\))
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 3694.50
The variables separate:
\( \displaystyle \frac{dP}{P} = 0.08\,dt \)
Integrating both sides:
\(\log P = 0.08t + c\)
\(P = P_0e^{0.08t}\)
Here \(P_0 = 500\), the amount at t = 0. At t = 25, \(0.08 \times 25 = 2\):
\(P = 500e^{2}\)
\(P = 500 \times 7.389 = 3694.5\)
So the amount is Rs. 3694.50.