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Element
X is a radioactive isotope such that every 12 years, its mass decreases by half. Given that the initial mass of a sample of Element
X is
20 grams, how much of the element would remain after 18 years, to the nearest whole number?
Answer:

Element $\mathrm{X}$ is a radioactive isotope such that every $12$ years, its mass decreases by half. Given that the initial mass of a sample of Element $\mathrm{X}$ is $\mathbf{2 0}$ grams, how much of the element would remain after $18$ years, to the nearest whole number? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. Element

\mathrm{X}

is a radioactive isotope such that every

12

years, its mass decreases by half. Given that the initial mass of a sample of Element

\mathrm{X}

is

\mathbf{2 0}

grams, how much of the element would remain after

18

years, to the nearest whole number?

\newline

Answer:

Determine half-lives passed: Determine the number of half-lives that have passed in $18$ years. $\newline$ Since the half-life of Element X is $12$ years, we divide the total time elapsed ( $18$ years) by the half-life ( $12$ years) to find the number of half-lives. $\newline$ Number of half-lives = $\frac{18 \text{ years}}{12 \text{ years}} = 1.5$ half-lives.
Calculate remaining mass: Calculate the remaining mass of Element X after $1.5$ half-lives. $\newline$ The initial mass is $20$ grams. After one half-life ( $12$ years), the mass would be halved, so after $1.5$ half-lives, it would be halved $1.5$ times. $\newline$ Remaining mass $= \frac{\text{Initial mass}}{2^{\text{Number of half-lives}}}$ $\newline$ Remaining mass $= \frac{20 \text{ grams}}{2^{1.5}}$
Perform calculation: Perform the calculation to find the remaining mass. $2^{1.5}$ is the same as the square root of $2^3$ , which is the square root of $8$ . Remaining mass = $20$ grams / $\sqrt{8}$ Remaining mass $\approx 20$ grams / $2.828$ Remaining mass $\approx 7.071$ grams
Round remaining mass: Round the remaining mass to the nearest whole number. $\newline$ Rounded remaining mass $\approx 7$ grams

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