QuestionJuly 24, 2025

The half-life of radium is 1690 years. If 8C ) grams are present now.how much will be present in 840 years? square grams (Do not round until the final answer. Then round to the nearest thousandth as needed.)

The half-life of radium is 1690 years. If 8C ) grams are present now.how much will be present in 840 years? square grams (Do not round until the final answer. Then round to the nearest thousandth as needed.)
The half-life of radium is 1690 years. If 8C ) grams are present now.how much will be present in 840 years? square grams (Do not round until the final answer. Then round to the nearest thousandth as needed.)

Solution
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Answer

56.64 grams Explanation 1. Determine the decay constant Use the formula for half-life: T_{1/2} = \frac{\ln(2)}{k}, where T_{1/2} = 1690 years. Solve for k: \[ k = \frac{\ln(2)}{1690} \] 2. Calculate remaining amount after 840 years Use the exponential decay formula: N(t) = N_0 e^{-kt}, where N_0 = 80 grams and t = 840 years. \[ N(840) = 80 \times e^{-\left(\frac{\ln(2)}{1690}\right) \times 840} \] 3. Compute the final amount Substitute values and compute: \[ N(840) = 80 \times e^{-\left(\frac{\ln(2)}{1690}\right) \times 840} \approx 80 \times e^{-0.344} \approx 80 \times 0.708 \]

Explanation

1. Determine the decay constant<br /> Use the formula for half-life: $T_{1/2} = \frac{\ln(2)}{k}$, where $T_{1/2} = 1690$ years. Solve for $k$: <br />\[ k = \frac{\ln(2)}{1690} \]<br /><br />2. Calculate remaining amount after 840 years<br /> Use the exponential decay formula: $N(t) = N_0 e^{-kt}$, where $N_0 = 80$ grams and $t = 840$ years.<br />\[ N(840) = 80 \times e^{-\left(\frac{\ln(2)}{1690}\right) \times 840} \]<br /><br />3. Compute the final amount<br /> Substitute values and compute:<br />\[ N(840) = 80 \times e^{-\left(\frac{\ln(2)}{1690}\right) \times 840} \approx 80 \times e^{-0.344} \approx 80 \times 0.708 \]
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