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**Pupil** A radioactive substance has a half-life of 20 days.

a) How much time is required so that only $\displaystyle \frac{1}{32}$ of the original amount remains?

b) Find the rate of decay at this time.

My attempts:

a) Since after 20 days the substance decays by half of its mass, after 40 days it decays to $\displaystyle {1}{4}$, and after 60 days it decays to $\displaystyle \frac{1}{8}$. After 80 days it decays to $\displaystyle \frac {1}{16}$ of its original mass and after 100 days it decays to $\displaystyle {1}{32}$ of its original mass. Therefore, it will take 100 days for the substance to decay to [tex]\frac{1}{32}/MATH] of its original mass.

b) $\displaystyle f(t) = \frac{1}{32}(\frac{1}{2})^{\frac{t}{20}$

$\displaystyle f'(t) =\frac{1}{32}(\frac{1}{2})^{\frac{t}{20}} × (ln\frac{1}{2}) × (\frac{1}{20}) $

$\displaystyle f'(100) =\frac{1}{32}(\frac{1}{2})^{\frac{100}{20}} × (ln\frac{1}{2}) × (\frac{1}{20}) $

$\displaystyle f'(100) = -3.384 × 10^{-5}$ is the rate of decay when $\displaystyle t = 100 $ days.

I just wanted to know if my answers are correct? I checked with derivative calculators and they have given me very different answers for b. Thanks in advance.