The half-life of a newly discovered radioactive element is 30 seconds. To the nearest tenth of a second, how long will it take for a sample

Question

The half-life of a newly discovered radioactive element is 30 seconds. To the nearest tenth of a second, how long will it take for a sample of 9 grams to decay to 0.72 grams

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Diễm Kiều 4 years 2021-08-03T19:55:38+00:00 1 Answers 33 views 0

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  1. thienthanh
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    2021-08-03T19:57:04+00:00
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    Answer:

    It will take about 109.3 seconds for nine grams of the element to decay to 0.72 grams.

    Step-by-step explanation:

    We can write a half-life function to model our function.

    A half-life function has the form:

    \displaystyle A=A_0\left(\frac{1}{2}\right)^{t/d}

    Where A₀ is the initial amount, t is the time that has passes (in this case seconds), d is the half-life, and A is the amount after t seconds.

    Since the half-life of the element is 30 seconds, d = 30. Our initial sample has nine grams, so A₀ is 9. Substitute:

    \displaystyle A=9\left(\frac{1}{2}\right)^{t/30}

    We want to find the time it will take for the element to decay to 0.72 grams. So, we can let A = 0.72 and solve for t:

    \displaystyle 0.72=9\left(\frac{1}{2}\right)^{t/30}

    Divide both sides by 9:

    \displaystyle 0.08=\left(\frac{1}{2}\right)^{t/30}

    We can take the natural log of both sides:

    \displaystyle \ln(0.08)=\ln\left(\left(\frac{1}{2}\right)^{t/30}\right)

    By logarithm properties:

    \displaystyle \ln(0.08)=\frac{t}{30}\ln(0.5)

    Solve for t:

    \displaystyle t=\frac{30\ln(0.08)}{\ln(0.5)}\approx109.3\text{ seconds}

    So, it will take about 109.3 seconds for nine grams of the element to decay to 0.72 grams.

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