Question

In a bank, principal increases continuously at the rate of r % per year. Find the value of r if Rs 100 doubles itself in 10 years (log­ e 2 = 0.6931).

Answer 1

Let principal, time and rate of the interest be p, t and r respectively. It is given that principal increases continuously at rate of r% per year.

dp dt =( r 100 )p dp p =( r 100 )dt

Integrate both sides.

∫ dp p = ∫ ( r 100 ) dt logp= r 100 ∫ dt logp= rt 100 +c p= e rt 100 +c (1)

It is given that when t=0, p=100.

100= e c (2)

Also when t=10,p=200.

200= e r 10 +c 200= e r 10 × e c 200= e r 10 ×100 e r 10 =2

Take log to both sides.

r 10 =log2 r 10 =0.6931 r=6.931%

Thus, the rate of the interest of the bank is 6.931%per annum.