# The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 4000 years?

${a}_{n} = 17.486 \text{ }$milligrams

#### Explanation:

The half life $= 1590 \text{ }$years

${t}_{0} = 100 \text{ }$time$= 0$
${t}_{1} = 50 \text{ }$time$= 1590$
${t}_{2} = 25 \text{ }$time$= 2 \left(1590\right)$
${t}_{3} = 12.5 \text{ }$time$= 3 \left(1590\right)$

${a}_{n} = {a}_{0} \cdot {\left(\frac{1}{2}\right)}^{n}$

$1 \text{ period"=1590" }$ years

$n = \frac{4000}{1590} = 2.51572327$

${a}_{n} = 100 \cdot {\left(\frac{1}{2}\right)}^{2.51572327}$

${a}_{n} = 17.486 \text{ }$milligrams

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