$$ N_{0}\left(\frac{1}{2}\right)= $$ 32) Rules of 70 and 72 : The rules state that it takes about $$ 70 / i $$ of $$ 72 / i $$ years for money to double at $$ i $$ percent, compounded continuously, using whichever of 70 or 72 is easier to divide by $$ i $$. a) Show that it takes $$ t=\ln 2 / r $$ years for money to double if it is invested at annual interest rate $$ r $$ (in decimal form) compounded continuously.

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Given the formula for continuously compounded interest:

$$
N(t) = N_0 e^{rt}
$$

we want the money to double, so we set $$ N(t) = 2N_0 $$:

$$
2N_0 = N_0 e^{rt}
$$

Dividing both sides by $$ N_0 $$:

$$
2 = e^{rt}
$$

Taking the natural logarithm of both sides yields:

$$
\ln 2 = rt
$$

Solving for $$ t $$:

$$
t = \frac{\ln 2}{r}
$$
It takes $$ t = \frac{\ln 2}{r} $$ years for money to double at an annual interest rate $$ r $$ when compounded continuously.

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