Exponential Growth and decay?

Money deposited in a bank that compounds continuously accumulates at a rate proportional to the amount present. Suppose that an initial deposit of P0 dollars doubles in 10 years.

1) Find the interest rate r that the bank is paying.

2) Determine the amount of money in the account at any time t.

3) How long will it take for an initial deposit of $1000 to grow to $5000?

Exponential Growth and decay?interest rate

use the formula

P(n) = P0 * (1+r)^n

where n is the number of years and r is the interest rate

-%26gt; P(10) = 2P0 (1+r)^10

-%26gt;1/2 = (1+r)^10

-%26gt;log(1/2)=1+r

-%26gt;r=log(1/2)-1

Exponential Growth and decay?

loan

for continuous compount interest:

A = A0*e^rt

r=rate, t=time

for an amount to double in 10 yrs:

e^(r*10) = 2

ln(e^(r*10)) = ln(2) , use ln(a^b) = b*ln(a)

(r*10) = ln(2)

r= ln(2)/10 = 0.0693, or 6.93%

2) A = P0e^0.0693t

3)

A = P0e^(0.0693t)

e^(0.0693t) = 5

ln(e^(0.0693t) = ln(5)

(0.0693t) = ln(5)

t = ln(5)/0.0693 = 23.22 yrs

note: remember that ln(e^x) = xln(e) = x*1 = x