Respuesta :
Answer:
1152
Step-by-step explanation:
given that in the formula
[tex]A = I (e^{kt)}[/tex]
half life is 1000 years
In other words when t =1000 A=l/2
[tex]A = I (e^{kt)}[/tex]
Since A = l/2 we get
[tex]0.5=e^{1000k} \\1000k = ln 0.5 = -0.6931\\k = -0.00069[/tex]
So equation is
[tex]A= le^{-0.00069t}[/tex]
When A/l = 0.45, we have
[tex]0.45=e^{-0.00069t} \\-0.00069t = -0.79851\\t=1152[/tex]
i.e. in 1152 years we expect to decay to 45%.
The age of the radioactive isotope can be determined from the amount
of the isotope remaining and the rate it decomposes.
Correct response:
- The number of years required is 1,152 years
Method used for the calculation of the time duration required
The function for the rate of decay for the radioactive isotope is; A = [tex]I \cdot e ^{k \cdot t}[/tex]
Where;
A = The amount of material left after time t
k = The rate
t = The time duration
I = The initial amount of the material at t = 0
The half life of the given isotope = 1,000
Therefore;
When t = 1,000, A = [tex]\dfrac{I}{2} [/tex], which gives;
[tex]\dfrac{I}{2} = \mathbf{I \cdot e^{1000 \cdot k}}[/tex]
[tex]\dfrac{1}{2} = e^{1000 \cdot k}[/tex]
[tex]\mathbf{ln\left(\dfrac{1}{2} \right)} = 1000 \cdot k[/tex]
[tex]k = \dfrac{ln\left(\frac{1}{2}\right) }{1000} \approx \mathbf{ -6.9315 \times 10^{-4}}[/tex]
When the isotope decays to 45%, we have;
[tex]45\% = 0.45 = e^{\left(-6.9315 \times 10^{-4} \right)\times t}[/tex]
Which gives;
-6.9315 × 10⁻⁴ × t = ㏑(0.45)
[tex]t \approx \mathbf{ \dfrac{ln(0.45)}{-6.9315 \times 10^{-4}}} \approx 1,152[/tex]
- The number of years required for the given amount of isotope to decay to 45% is t ≈ 1,152 years
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