QCE Maths Methods IA1 strategy: 2026 guide

Step 1. Formulate, define the problem

A patient is prescribed a 200 mg dose of paracetamol every 6 hours. Predict the steady-state concentration and the time to reach it.

Step 2. Formulate, state assumptions

(i) The drug is absorbed instantaneously into the bloodstream. (ii) Elimination follows first-order kinetics with half-life $2$ hours. (iii) The volume of distribution is constant. Each assumption is plausible at therapeutic doses for healthy adults; they will be revisited in the evaluation.

Step 3. Formulate, identify variables

Time $t$ (hours). Concentration $C(t)$ (in dose-equivalent mg units). Decay constant $k = \ln 2 / 2 \approx 0.347$ per hour.

Step 4. Solve, between-dose decay

First-order decay: $C(t) = C_0 e^{-kt}$. Over one $6$-hour interval the multiplier is $r = e^{-6k} = e^{-2.08} \approx 0.125$.

Step 5. Solve, sum the geometric series

Just before the $n$th dose, the accumulated concentration is the partial sum of a geometric series:

Step 6. Solve, take the limit for steady state

As $n \to \infty$, $r^n \to 0$ (since $|r| < 1$), giving steady-state $C_\infty = \frac{C_0}{1 - r} = \frac{200}{1 - 0.125} \approx 229$ dose-equivalent mg.

Step 7. Solve, time to 95 percent of steady state

Require $r^n < 0.05$. Taking logs: $n > \frac{\ln 0.05}{\ln 0.125} \approx 1.44$. So $n = 2$ doses ($12$ hours) brings the patient within $5\%$ of steady state.

Step 8. Evaluate and verify in context

Within $12$ hours of the first dose, concentration is close to its long-run value. Check the limit: $C_\infty \approx 1.14 \times C_0$, sensible because $r \ll 1$ (most drug has cleared by the next dose).

Step 9. Evaluate, identify limitations

Real absorption is not instantaneous, so the model overstates the peak. Constant elimination is reasonable at therapeutic dose. Volume of distribution varies between individuals (about $0.9$ L/kg of body mass).

Step 10. Refine

Replace instantaneous absorption with a single-compartment model with absorption rate constant $k_a$, giving $C(t) = \frac{F D k_a}{V (k_a - k)} (e^{-kt} - e^{-k_a t})$. Re-solve numerically and compare peak and steady-state values. Use graphing technology to plot both models on one axis.

Step 11. Communicate

Report sections: Introduction, Formulate, Solve, Evaluate and verify, Refine, Conclusion, Appendix. Each figure labelled with axes, units, source; each table with headers and consistent significant figures.

Answer

Steady-state concentration $\approx 229$ dose-equivalent mg; $95\%$ of steady state reached within $12$ hours (two dosing intervals).

Step 1. Formulate, define the problem

A manufacturer wants to make a closed cylindrical can of volume $500$ cm$^3$ using as little sheet metal as possible. Find the radius and height that minimise surface area, and report the minimum surface area.

Step 2. Formulate, identify variables

Radius $r$ (cm), height $h$ (cm), volume $V = 500$ cm$^3$, surface area $S$ (cm$^2$).

Step 3. Formulate, assumptions

The can is a right circular cylinder with rigid wall (sheet thickness ignored), the lid and base are flat circular discs, no seam allowance.

Step 4. Formulate, constraint

$V = \pi r^2 h = 500$, so $h = \frac{500}{\pi r^2}$.

Step 5. Formulate, objective in one variable

Closed cylinder surface area:

Step 6. Solve, differentiate.

Step 7. Solve, stationary point

$S'(r) = 0$ gives $4 \pi r^3 = 1000$, so $r^3 = \frac{250}{\pi}$ and $r = \left(\frac{250}{\pi}\right)^{1/3} \approx 4.30$ cm.

Step 8. Solve, second-derivative test

$S''(r) = 4 \pi + \frac{2000}{r^3} > 0$ for $r > 0$, confirming a local minimum (and the only stationary point on $r > 0$, so a global minimum).

Step 9. Solve, compute $h$ and $S$

$h = \frac{500}{\pi (4.30)^2} \approx 8.60$ cm. $S(4.30) \approx 2 \pi (4.30)^2 + \frac{1000}{4.30} \approx 348$ cm$^2$.

Step 10. Evaluate and verify

Note $h = 2r$: the optimal closed cylinder has height equal to diameter. This is a known textbook result and supports reasonableness of the answer.

Step 11. Evaluate, limitations

Real cans include a seam allowance and rolled lip; lids are usually thinner sheet than walls. A refined model could weight wall and lid metal differently.

Step 12. Communicate

Present a sketch of $S$ versus $r$ over $r \in (0, 10]$ showing the minimum; state results to three significant figures with units.

Answer

$r \approx 4.30$ cm, $h \approx 8.60$ cm, $S \approx 348$ cm$^2$.

Step 1. Formulate, define the problem

The water depth $D$ (metres) at a wharf is modelled as $D(t) = 5 + 2 \sin\!\left(\frac{\pi t}{6}\right)$ where $t$ is hours after midnight. A boat needs at least $6$ metres of water. For what fraction of a $24$-hour day can the boat enter?

Step 2. Formulate, identify model parameters

Mean depth $5$ m, amplitude $2$ m, period $T = 2 \pi / (\pi / 6) = 12$ hours (semi-diurnal tide), so two high tides per day.

Step 3. Formulate, assumptions

Tidal motion is purely sinusoidal (no weather effect, no wind, no seasonal range change). The instrument reads to $\pm 0.05$ m.

Step 4. Solve, set up the inequality

Need $D(t) \ge 6$, i.e. $5 + 2 \sin\!\left(\frac{\pi t}{6}\right) \ge 6$, so $\sin\!\left(\frac{\pi t}{6}\right) \ge \frac{1}{2}$.

Step 5. Solve, find boundary times within one period

$\sin \theta = 1/2$ at $\theta = \pi/6$ and $\theta = 5 \pi / 6$. With $\theta = \pi t / 6$, this gives $t = 1$ and $t = 5$.

Step 6. Solve, find admissible interval in one period

$\sin \theta \ge 1/2$ for $\theta \in [\pi/6, 5\pi/6]$, hence $t \in [1, 5]$ hours each period.

Step 7. Solve, total in 24 hours

Two periods per day, each contributing $5 - 1 = 4$ hours of access. Total: $8$ hours.

Step 8. Evaluate and verify

As a fraction of the day: $8 / 24 = 1/3 \approx 33.3\%$. Sanity check: the model exceeds $6$ m on the upper third of its range, so a third-of-cycle access window is consistent.

Step 9. Evaluate, limitations

Real tides are influenced by weather (storm surge) and lunar cycle. The model uses a single fixed amplitude.

Step 10. Refine

Replace constant amplitude with $A(t) = 2 + 0.3 \sin(2 \pi t / (14 \times 24))$ to capture the spring-neap fortnightly variation, and re-solve numerically.

Step 11. Communicate

Plot $D(t)$ over $t \in [0, 24]$ with the $6$ m threshold marked; shade the admissible windows.

Answer

The boat can enter for about $8$ hours per day, roughly one-third of the time.

Q1

(1) Define the problem, (2) make assumptions, (3) identify variables, (4) formulate the model, (5) solve, (6) interpret, (7) evaluate, (8) refine.

Q2

Formulate (translate context into a mathematical model with justified assumptions); Solve (apply correct mathematics with appropriate technology); Evaluate and verify (assess reasonableness and refine); Communicate (organise the report and use mathematical notation correctly).

Q3

Optimum occurs at $h = 2 r$ (height equals diameter), i.e. $h : r = 2 : 1$. It is a known textbook result: a quick reasonableness check that the calculation was set up correctly.

Q4

Logistic growth is fastest at the inflection point, where $P = L / 2 = 500$. Justification: the per-capita rate $\frac{dP}{dt} = k P (1 - P/L)$ is a downward parabola in $P$, maximised at $P = L/2$.

Q5

Solve $5 + 2 \sin(\pi t / 6) = 4.5$, i.e. $\sin(\pi t / 6) = -1/4$, on $t \in [0, 24)$.

Q6

Replace the instantaneous-absorption assumption with a first-order absorption phase characterised by rate constant $k_a$, giving the two-exponential single-compartment model $C(t) = \frac{F D k_a}{V (k_a - k)} (e^{-kt} - e^{-k_a t})$.

Q7

Example: assume daily traffic flow is periodic with period $24$ hours. Justification: morning and evening commuting peaks repeat each weekday, broadly consistent with sinusoidal behaviour; the model would not capture weekend or holiday patterns.

Q8

Strong structure: (1) Introduction (problem statement and approach), (2) Formulate (variables, assumptions with justifications), (3) Solve (mathematical work with technology evidence), (4) Evaluate and verify (sensitivity, alternative scenarios, limitations), (5) Refine (modified model, comparison), (6) Conclusion (direct answer with units), (7) Appendix (calculator or software screenshots, supplementary calculations), (8) References.