Tritium, a radioactive isotope of hydrogen, is often used in emergency EXIT signs. Because of the radioactive substance present in these signs, proper disposal is a matter of concern to the Environmental Protection Agency. The half-life of tritium is approximately $12$ years. That is, every $12$ years, the amount of tritium decreases by $50\%$ . If a new tritium EXIT sign contains $25$ curies of tritium, approximately how many curies of tritium will remain after $24$ years? $\newline$ Choose $1$ answer: $\newline$ (A) $6.25$ curies $\newline$ (B) $12.5$ curies $\newline$ (C) $25$ curies $\newline$ (D) $30$ curies

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Exponential growth and decay: word problems

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Q. Tritium, a radioactive isotope of hydrogen, is often used in emergency EXIT signs. Because of the radioactive substance present in these signs, proper disposal is a matter of concern to the Environmental Protection Agency. The half-life of tritium is approximately

12

years. That is, every

12

years, the amount of tritium decreases by

50\%

. If a new tritium EXIT sign contains

25

curies of tritium, approximately how many curies of tritium will remain after

24

years?

\newline

Choose

1

answer:

\newline

(A)

6.25

curies

\newline

(B)

12.5

curies

\newline

(C)

25

curies

\newline

(D)

30

curies

Identify Values: Identify the values of the initial amount $a$ , total time $t$ , and half-life period $h$ . $\newline$ Initial amount $a$ = $25$ curies $\newline$ Total time $t$ = $24$ years $\newline$ Half-life period $h$ = $12$ years
Use Half-life Formula: Use the half-life formula to calculate the remaining amount of tritium after $24$ years. $\newline$ The formula is $y = a \times (1/2)^{(t/h)}$ , where $y$ is the remaining amount after time $t$ , $a$ is the initial amount, and $h$ is the half-life period.
Substitute Values: Substitute the values into the formula. $\newline$ $y = 25 \times \left(\frac{1}{2}\right)^{\frac{24}{12}}$
Simplify Exponent: Simplify the exponent in the formula. $\frac{24}{12} = 2$ , so the exponent becomes $(\frac{1}{2})^2$ .
Calculate Remaining Quantity: Calculate the remaining quantity of tritium. $\newline$ $y = 25 \times (\frac{1}{2})^2$ $\newline$ $y = 25 \times \frac{1}{4}$ $\newline$ $y = 6.25$