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Question

In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs1000 double itself?

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Solution

It is given that
Rate of change of principal = 5% of Principal

dPdt=5%×P

(Where P is Principal at any time t)

dPdt=5100×P

dPdt=P20

dpp=dt20

Integrating both sides
We get,

dpp=dt20

log|P|=t2+c

P=et20+c

P=et20×ec

Putting ec=r
P=r.et20 (i)

Now, we need to find the time (t) when principal is double Rs2000

Timet=0t=?Principal10002×1000=2000

At t=0,P=1000

Putting in (i) then we get,

1000=r.e020

1000=r.1

putting r=1000 in (i) then we get,

P=1000.et20

Now,
we need to find time t when Principal is
Rs2000 i.e.,
P=2000, t=t

20001000=et20

et20=2

Integrating both sides
We get,

loge et20=loge2

t2logee=loge2

(logxn=nlogx)

t20×1=loge2 (loge=1)

t=20loge 2

Final Answer:
Hence, the required time is

t=20loge 2



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