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A scientist is researching the effects of caffeine - AQA - A-Level Maths Pure - Question 10 - 2018 - Paper 1

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A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using $$m = m_0 e^{-kt}$$ where $m_0$ milligrams is the initial mas... show full transcript

Worked Solution & Example Answer:A scientist is researching the effects of caffeine - AQA - A-Level Maths Pure - Question 10 - 2018 - Paper 1

Step 1

Use the model to form an equation to find $k$ with 5.7 hours: $m = \frac{1}{2} m_0$

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Answer

Since it takes 5.7 hours for the caffeine to halve: m=12m0m = \frac{1}{2} m_0 Substituting into the model gives: 12m0=m0e−5.7k\frac{1}{2} m_0 = m_0 e^{-5.7k} Dividing both sides by m0m_0 (assuming m0≠0m_0 \neq 0) leads to: 12=e−5.7k\frac{1}{2} = e^{-5.7k} Taking the natural logarithm: ln⁡(12)=−5.7k\ln\left(\frac{1}{2}\right) = -5.7k Therefore: k=−ln⁡(12)5.7≈0.1216047k = -\frac{\ln\left(\frac{1}{2}\right)}{5.7} \approx 0.1216047

Step 2

Use $m = m_0 e^{-kt}$ with $m_0 = 400$ and $k \approx 0.1216$ to find $m$ at $t = 4$

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Answer

At midday (4 hours after 8 am), we substitute m0=400m_0 = 400 mg (2 cups of coffee) and k≈0.1216k \approx 0.1216 into the model: m=400e−0.1216×4m = 400 e^{-0.1216 \times 4} Calculating this gives: m≈400×e−0.4864≈400×0.61557≈246.23m \approx 400 \times e^{-0.4864} \approx 400 \times 0.61557 \approx 246.23 Thus, the estimated mass of caffeine in the scientist's body at midday is approximately 246 mg.

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