QuestionJune 4, 2025

7. A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years. a. How much of the sample will remain after 6 years? 3 years? 1 year? square After 6 years... After 3 years... b. Find a function f which models the amount of radium f(t) in mg, remaining after t years. f(t)=

7. A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years. a. How much of the sample will remain after 6 years? 3 years? 1 year? square After 6 years... After 3 years... b. Find a function f which models the amount of radium f(t) in mg, remaining after t years. f(t)=
7. A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years. a. How much of the sample will remain after 6 years? 3 years? 1 year? square After 6 years... After 3 years... b. Find a function f which models the amount of radium f(t) in mg, remaining after t years. f(t)=

Solution
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Answer

After 6 years: 0.75 mg; After 3 years: 1.06 mg; After 1 year: 1.34 mg. ### f(t) = 1.5 \times \left(\frac{1}{2}\right)^{\frac{t}{6}} Explanation 1. Calculate remaining radium after 6 years Use the half-life formula: After one half-life (6 years), half of the sample remains. 1.5 \, \text{mg} \times \frac{1}{2} = 0.75 \, \text{mg}. 2. Calculate remaining radium after 3 years Use the decay formula: A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}, where A_0 = 1.5 \, \text{mg}, t = 3, and T = 6. A = 1.5 \times \left(\frac{1}{2}\right)^{\frac{3}{6}} = 1.5 \times \left(\frac{1}{2}\right)^{0.5} \approx 1.06 \, \text{mg}. 3. Calculate remaining radium after 1 year Use the decay formula again: A = 1.5 \times \left(\frac{1}{2}\right)^{\frac{1}{6}} \approx 1.34 \, \text{mg}. 4. Define the function for remaining radium The function is f(t) = 1.5 \times \left(\frac{1}{2}\right)^{\frac{t}{6}}.

Explanation

1. Calculate remaining radium after 6 years<br /> Use the half-life formula: After one half-life (6 years), half of the sample remains. $1.5 \, \text{mg} \times \frac{1}{2} = 0.75 \, \text{mg}$.<br /><br />2. Calculate remaining radium after 3 years<br /> Use the decay formula: $A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$, where $A_0 = 1.5 \, \text{mg}$, $t = 3$, and $T = 6$. $A = 1.5 \times \left(\frac{1}{2}\right)^{\frac{3}{6}} = 1.5 \times \left(\frac{1}{2}\right)^{0.5} \approx 1.06 \, \text{mg}$.<br /><br />3. Calculate remaining radium after 1 year<br /> Use the decay formula again: $A = 1.5 \times \left(\frac{1}{2}\right)^{\frac{1}{6}} \approx 1.34 \, \text{mg}$.<br /><br />4. Define the function for remaining radium<br /> The function is $f(t) = 1.5 \times \left(\frac{1}{2}\right)^{\frac{t}{6}}$.
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