Expression for the radioactive decay and remaining amount after 8 days will be represented by option (3).
Expression for the exponential decay of a radioactive element is given by,
[tex]A(t)=A_0(1-r)^{t}[/tex]
Here, [tex]A(t)=[/tex] Final amount
[tex]A_0=[/tex] Initial amount
[tex]r=[/tex] Rate of decay
[tex]t=[/tex] Time or period
For a radioactive element,
- Initial mass = 49 kg
- Half life = 42 days
After 42 days remaining mass of the element is half of the initial amount.
From the given expression,
[tex]\frac{49}{2}=49(1-r)^{42}[/tex]
[tex](\frac{1}{2})^{\frac{1}{42}}=1-r[/tex]
[tex]r=1-(\frac{1}{2})^{\frac{1}{42} }[/tex]
By substituting the value of 'r' in the expression,
[tex]A(t)=49[1-(1-(\frac{1}{2})^{\frac{1}{42}})]^t[/tex]
[tex]y=49(\frac{1}{2})^{\frac{1}{42}t}[/tex]
For t = 8 days,
[tex]y=49(\frac{1}{2})^{\frac{1}{42}\times 8}[/tex]
[tex]y=49(\frac{1}{2})^{\frac{4}{21} }[/tex]
[tex]y=42.940[/tex] kg
Therefore, expression for the decay of this material will be [tex]y=49(\frac{1}{2})^{\frac{1}{42}t}[/tex] and amount remaining after 8 days will be 42.940 kg.
Option (3) will be the correct option.
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