Q9 of 462 Page 22

A ban pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.

[Take e0.08≈1.0833]

Let the principal, rate and time be Rs P, r and t years.

Also, let the initial principal be Po.



⇒


Integrating both sides, we have


⇒ ∫∫dt


⇒ log|P| = t + c……(1)


Now, at t = 0, P = Po


log| Po | = 0 + c


⇒ c = log| Po |……(2)


Putting the value of c in equation (1) we have,


log|P| = t + log|Po|


⇒ log|P| – log|Po| = t


⇒ (log |P| – log|Po|) = t []


⇒ log ( = t ……(3)


Now, t = 1 year, r = 8%


∴ log ( = ×1


⇒ log ( = 0.08


⇒


⇒


⇒


(Given: = 1.0833)


⇒


∴ Percentage increase = 0.0833×100 = 8.33%


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