QuestionMay 3, 2025

The half-life of a radioactive substance is 3 days and the initial (original) amount was 16 mg. What is the mass 7 weeks after the start? square Note: Make sure to change weeks to days

The half-life of a radioactive substance is 3 days and the initial (original) amount was 16 mg. What is the mass 7 weeks after the start? square Note: Make sure to change weeks to days
The half-life of a radioactive substance is 3 days and the initial (original) amount was 16 mg. What is the mass 7 weeks after the start? square Note: Make sure to change weeks to days

Solution
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Answer

Approximately 0.0001728 mg Explanation 1. Convert weeks to days 7 weeks = 7 × 7 = 49 days. 2. Use the half-life formula The formula for decay is N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}, where N_0 is the initial amount, t is time, and T_{1/2} is the half-life. 3. Calculate remaining mass Substitute N_0 = 16 mg, t = 49 days, and T_{1/2} = 3 days into the formula: N(49) = 16 \left(\frac{1}{2}\right)^{\frac{49}{3}}. Calculate the exponent: \frac{49}{3} \approx 16.3333. Compute N(49) = 16 \times \left(\frac{1}{2}\right)^{16.3333}. Approximate N(49) \approx 16 \times 0.0000108 \approx 0.0001728 mg.

Explanation

1. Convert weeks to days<br /> 7 weeks = 7 × 7 = 49 days.<br />2. Use the half-life formula<br /> The formula for decay is $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, where $N_0$ is the initial amount, $t$ is time, and $T_{1/2}$ is the half-life.<br />3. Calculate remaining mass<br /> Substitute $N_0 = 16$ mg, $t = 49$ days, and $T_{1/2} = 3$ days into the formula: <br /> $N(49) = 16 \left(\frac{1}{2}\right)^{\frac{49}{3}}$.<br /> Calculate the exponent: $\frac{49}{3} \approx 16.3333$.<br /> Compute $N(49) = 16 \times \left(\frac{1}{2}\right)^{16.3333}$.<br /> Approximate $N(49) \approx 16 \times 0.0000108 \approx 0.0001728$ mg.
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