A-Level AQA Maths Pure Exponential & Logarithms: It takes 15.9 hours for half of

It takes 15.9 hours for half of the sodium atoms to decay - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 3

Question 5

It takes 15.9 hours for half of the sodium atoms to decay. Determine the number of days required for at least 90% of the number of atoms in the original sample to d... show full transcript

Worked Solution & Example Answer:It takes 15.9 hours for half of the sodium atoms to decay - 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, we can substitute the known values into the equation:

$N = N_0 e^{-kt}$

Substituting:

$\frac{N_0}{2} = N_0 e^{-k(15.9)}$

Dividing both sides by $N_0$ gives:

$\frac{1}{2} = e^{-k(15.9)}$

Taking the natural logarithm of both sides:

$\ln\left(\frac{1}{2}\right) = -k(15.9)$

Solving for $k$ :

$k = -\frac{\ln\left(\frac{1}{2}\right)}{15.9} = 0.0436$

Step 2

Substitutes their value of $k$ and $N = 0.1N_0$ in the model to find $t$

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Answer

Next, we want to find the time $t$ when $90\%$ has decayed. Therefore, we set:

$N = 0.1N_0$

Using the model again:

$0.1N_0 = N_0 e^{-kt}$

Dividing by $N_0$ :

$0.1 = e^{-kt}$

Taking the natural logarithm:

$\ln(0.1) = -kt$

Using the value of $k$ :

$t = -\frac{\ln(0.1)}{0.0436}$

Step 3

Solves their equation correctly to find $t$

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Answer

Calculating:

$t = -\frac{-2.3026}{0.0436} \approx 52.3 \text{ hours}$

To convert hours to days:

$t_{days} = \frac{52.3}{24} \approx 2.18 \text{ days}$

Rounding appropriately yields:

$t_{days} \approx 2.2 \text{ days}$

It takes 15.9 hours for half of the sodium atoms to decay - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 3

Worked Solution & Example Answer:It takes 15.9 hours for half of the sodium atoms to decay - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 3

Substitutes $t = 15.9$ hours and $N = \frac{N_0}{2}$ in the model to find $k$

Substitutes their value of $k$ and $N = 0.1N_0$ in the model to find $t$

Solves their equation correctly to find $t$

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