Answer:
Initial mass: 64 mg
Mass of the sample 4 weeks after the start: [tex] \bold{\sf \boxed{0.5} }[/tex] mg
Step-by-step explanation:
To solve these questions, we will use the exponential decay formula for radioactive decay, which can be expressed as:
[tex]\large\boxed{\boxed{ \sf m(t) = m_0 \left(\dfrac{1}{2}\right)^{\frac{t}{T_{\frac{1}{2}}}} }}[/tex]
where:
- [tex] \bold{\sf m(t) }[/tex] is the mass of the substance at time [tex] \bold{\sf t }[/tex],
- [tex] \bold{\sf m_0 }[/tex] is the initial mass of the substance,
- [tex] \bold{\sf T_{1/2} }[/tex] is the half-life of the substance,
- [tex] \bold{\sf t }[/tex] is the elapsed time.
Finding the initial mass [tex] \bold{\sf ( m_0 )}[/tex]
- Given:
- Half-life [tex] \bold{\sf T_{1/2} = 4 }[/tex] days
- Mass after 16 days [tex] \bold{\sf m(16) = 4 }[/tex] mg
Using the exponential decay formula:
[tex]\sf m(16) = m_0 \left(\dfrac{1}{2}\right)^{\frac{16}{4}}[/tex]
[tex] \sf 4 = m_0 \left(\dfrac{1}{2}\right)^4 [/tex]
[tex] \sf 4 = m_0 \left(\dfrac{1}{16}\right) [/tex]
[tex] \sf m_0 = 4 \times 16 [/tex]
[tex] \sf m_0 = 64 \textsf{ mg} [/tex]
Therefore, the initial mass [tex] \bold{\sf ( m_0 )}[/tex] of the sample was [tex] \bold{\sf \boxed{64} }[/tex] mg.
\[tex]\dotfill[/tex]
Finding the mass 4 weeks (7 × 4 = 28 days) after the start:
To find the mass after 28 days (4 weeks), substitute [tex] \bold{\sf t = 28 }[/tex] days into the exponential decay formula:
[tex]\sf m(28) = 64 \left(\dfrac{1}{2}\right)^{\frac{28}{4}}[/tex]
[tex] \sf m(28) = 64 \left(\dfrac{1}{2}\right)^7 [/tex]
[tex] \sf m(28) = 64 \times \dfrac{1}{128} [/tex]
[tex] \sf m(28) = \dfrac{64}{128} [/tex]
[tex] \sf m(28) = 0.5 \textsf{ mg} [/tex]
Therefore, the mass of the sample 4 weeks (28 days) after the start was [tex] \bold{\sf \boxed{0.5} }[/tex] mg.
