The number of radioactive atoms, $N$, in a sample of a sodium isotope after time $t$ hours can be modelled by
$$N = N_0 e^{-kt}$$
where $N_0$ is the initial number of radioactive atoms in the sample and $k$ is a positive constant - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 3
Question 5
The number of radioactive atoms, $N$, in a sample of a sodium isotope after time $t$ hours can be modelled by
$$N = N_0 e^{-kt}$$
where $N_0$ is the initial number... show full transcript
Worked Solution & Example Answer:The number of radioactive atoms, $N$, in a sample of a sodium isotope after time $t$ hours can be modelled by
$$N = N_0 e^{-kt}$$
where $N_0$ is the initial number of radioactive atoms in the sample and $k$ is a positive constant - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 3
Step 1
Substitutes $t = 15.9$ hours and $N = \frac{N_0}{2}$ in the model to find $k$
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Answer
Using the model:
2N0=N0e−k⋅15.9
Dividing both sides by N0 (assuming N0>0) gives:
21=e−k⋅15.9
Taking the natural logarithm:
−k⋅15.9=ln(21)
Thus,
k=15.9−ln(0.5)
Calculating yields:
k≈0.0436
Step 2
Substitutes their value of $k$ and $N = 0.1N_0$ in the model to find $t$
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Answer
Using:
0.1N0=N0e−0.0436t
Dividing both sides by N0 gives:
0.1=e−0.0436t
Taking the natural logarithm:
−0.0436t=ln(0.1)
Thus,
t=0.0436−ln(0.1)
Calculating this results in:
t≈52.3 hours
Converting hours to days:
2452.3≈2.18 days⇒2.2 days (to 2 decimal places)
