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Tritium is a radioactive isotope of hydrogen that decays by about $5\%$ per year. A large bottle of water that contained $450,000$ tritium atoms remained undisturbed for $11$ years. How much tritium does the bottle contain now? If necessary, round your answer to the nearest whole number. $\newline$ ____ tritium atoms $\newline$

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Exponential growth and decay: word problems

Full solution

Q. Tritium is a radioactive isotope of hydrogen that decays by about

5\%

per year. A large bottle of water that contained

450,000

tritium atoms remained undisturbed for

11

years. How much tritium does the bottle contain now? If necessary, round your answer to the nearest whole number.

\newline

____ tritium atoms

\newline

Determine Decay Type: Determine the type of decay process. Tritium decays by $5\%$ per year, which indicates an exponential decay process.
Identify Initial Amount: Identify the initial amount $a$ and the decay rate $r$ . The initial amount of tritium atoms is $a = 450,000$ . The decay rate per year is $r = 5\%$ or $0.05$ when expressed as a decimal.
Calculate Remaining Amount: Calculate the remaining amount of tritium after $11$ years using the exponential decay formula. $\newline$ The formula for exponential decay is $P(t) = a(1 - r)^t$ , where $P(t)$ is the amount of substance remaining after time $t$ , $a$ is the initial amount, $r$ is the decay rate, and $t$ is the time in years.
Substitute Values: Substitute the known values into the exponential decay formula. $P(11) = 450,000(1 - 0.05)^{11}$
Calculate Remaining Amount: Calculate the remaining amount of tritium. $\newline$ $P(11) = 450,000(0.95)^{11}$ $\newline$ $P(11) \approx 450,000(0.571753)^{11}$ $\newline$ $P(11) \approx 450,000 \times 0.571753$ $\newline$ $P(11) \approx 257,288.85$
Round Answer: Round the answer to the nearest whole number. $\newline$ The remaining amount of tritium is approximately $257,289$ atoms.

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