A Level · Cambridge (CIE) · 9709

Model exponential growth decay and threshold times

Calibrate an exponential model from two observations, find a threshold time and distinguish positive decay from reaching zero.

Section 3.8 · Differential equations

Pure Mathematics 3 · Differential equations · 3.8

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.8. Paper 1 calculus and Paper 3 Integration are assumed. This pathway uses first-order separable equations, including Paper 3 integration methods, with all context assumptions supplied. Second-order equations and integrating-factor methods are beyond this mapped requirement. Check factors before dividing, retain equilibrium solutions, and state the interval and physical meaning of an initial-value solution. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Solve y′ = ky and use initial conditions.
Use logarithms to solve exponential equations.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

If e^(2k) = 3/2, which value is k?

What you will learn

Solve and calibrate a proportional growth or decay equation.
Interpret its constant, doubling or halving time and finite-time predictions.

Use observations to calibrate the constant

The law dP/dt = kP has family P = Ae^(kt). A starting value P(0) = P₀ gives A = P₀. If P(a) = P₁ with positive amounts, then k = ln(P₁/P₀)/a. The ratio must use matching units and the elapsed interval a. Growth gives k > 0 and decay gives k < 0. In a supplied population model with P₀ = 400 and P(2) = 600, k = (1/2)ln(3/2) per hour.

This model is P(t) = 400(3/2)^(t/2), and its doubling time is ln 2/k = 2ln 2/ln(3/2), approximately 3.42 hours. The doubling time is a duration, whereas k is a reciprocal-time constant. Changing from hours to minutes changes the numerical value of k. Different time units do not produce different physical predictions when converted consistently.

A model has assumptions as well as a formula

For M′ = −λM with λ > 0, M = M₀e^(−λt). Its half-life is ln 2/λ. Starting from a positive M₀, this exponential stays positive at every finite t and approaches zero as t grows without bound. A practical detection threshold can be reached in finite time; exact zero cannot under this law. Solve the particular threshold rather than replacing it with a half-life.

Calibration to two observations does not establish that the rate law will remain valid indefinitely. State that predictions depend on a constant proportional rate and the supplied conditions continuing to hold. A continuously modelled population need not give an integer prediction; use the rounding requested by the problem and do not turn an approximate model value into a guaranteed count. Retain exact logarithms until the final numerical result.

Pause and explain

Under positive exponential decay, is exact zero reached at a finite time?

Put the idea to work

Worked example

A supplied population model has P′ = kP, P(0) = 400 and P(2) = 600, with time in hours. Find P(t) and its doubling time.

Show the worked solution

The starting value gives P = 400e^(kt). Use e^(2k) = 3/2 to obtain k = ln(3/2)/2.
Thus P(t) = 400(3/2)^(t/2).
Doubling requires e^(kt) = 2, so t = 2ln 2/ln(3/2) ≈ 3.42 hours.

Answer P(t) = 400(3/2)^(t/2); doubling time 2ln 2/ln(3/2) ≈ 3.42 h.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

A supplied decay model starts at 80 g and has 40 g after 3 hours. Find M(t), its positive decay constant λ and M(9).

Give me a hint

The amount halves in three hours.

Compare my reasoning

Use M = 80e^(−λt) and e^(−3λ) = 1/2.
Thus λ = ln 2/3 per hour and M(t) = 80·2^(−t/3).
After 9 hours there have been three halvings, so M(9) = 10 g.

λ = ln 2/3 h⁻¹; M(t) = 80·2^(−t/3); M(9) = 10 g.

Look for these in your work

Use a positive λ with the negative sign in the rate law.
Check the initial amount and repeated-halving prediction.

2 · Independent

P′ = kP, P(0) = 150 and P(4) = 240, with time in hours. Find k and the time when P first reaches 300.

Give me a hint

Use the ratio 240/150 before solving the doubling condition.

Compare my reasoning

The ratio gives e^(4k) = 8/5, hence k = ln(8/5)/4.
Reaching 300 requires e^(kt) = 2.
Thus t = 4ln 2/ln(8/5), approximately 5.90 hours, under the unchanged law.

k = ln(8/5)/4 h⁻¹; threshold time 4ln 2/ln(8/5) ≈ 5.90 h.

Look for these in your work

Calibrate using the actual elapsed interval.
Retain exact logarithms until final rounding.

3 · Transfer

A measured signal follows a supplied exponential-decay law. It starts at 90 units and is 30 after 2 hours. Find the first time it reaches 1 unit, and explain whether the model then predicts zero.

Give me a hint

Calibrate the decay from a factor of one third over two hours.

Compare my reasoning

The solution is q(t) = 90·3^(−t/2), with positive decay constant ln 3/2.
Set q = 1 to obtain t = 2ln 90/ln 3, approximately 8.19 hours.
The amount remains positive afterwards at every finite time; reaching the threshold is not reaching zero, and the prediction assumes the supplied law continues.

q(t) = 90·3^(−t/2); reaches 1 at 2ln 90/ln 3 ≈ 8.19 h; exact zero is not reached at finite time.

Look for these in your work

Distinguish the requested threshold from a halving time.
State the conditional interpretation and the limiting rather than finite zero.

Common mistakes

The logarithm uses the later-to-earlier quantity ratio.
A half-life is a duration and is not the decay constant itself.
Positive exponential decay approaches zero without reaching it at finite time.

Recall without your notes

What can you explain now?

For M′ = −λM with λ > 0, what is the half-life?

Compare with the explanation

ln 2/λ.

The ratio M(t)/M(0) is e^(−λt); set it to 1/2 and solve for the duration.

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Make it stick

Calibrate a proportional-rate model, solve an exact threshold equation and explain the difference between a constant, a doubling or halving duration, and a limiting value.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.