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Exponential growth and decay

Solve and interpret exponential growth and decay models of the form dNdt=kN\frac{dN}{dt} = kN and dNdt=k(N−P)\frac{dN}{dt} = k(N-P); apply Newton's law of cooling; use logarithms to find unknown times and rates.

Worked examples

Basic growth: finding a future value

Straightforward

Problem

A bacterial culture starts with 300 cells and grows at a rate satisfying dNdt=0.25N\dfrac{dN}{dt} = 0.25N. How many cells are present after 8 hours?

Finding kk from two data points, then predicting

Moderate

Problem

A radioactive substance has 600 g initially and 400 g after 5 years. Find the mass after 12 years, to the nearest gram.

Newton's law of cooling

Challenging

Problem

A cup of coffee at 95°C is placed in a 20°C room. After 5 minutes its temperature is 65°C. Find the temperature after 15 minutes. (Newton's law of cooling: dTdt=−k(T−Ts)\dfrac{dT}{dt} = -k(T - T_s), where TsT_s is the surrounding temperature.)

Practise

Q1·Straightforward
A population NN grows according to dNdt=0.4N\dfrac{dN}{dt} = 0.4N, with N(0)=250N(0) = 250. Find NN when t=3t = 3. Round your answer to the nearest whole number.
Q2·Straightforward
A radioactive substance decays so that dMdt=−0.05M\dfrac{dM}{dt} = -0.05M, with M(0)=8000M(0) = 8000 g. Find MM when t=20t = 20. Round your answer to the nearest whole number.
Q3·Straightforward
A quantity QQ satisfies dQdt=kQ\dfrac{dQ}{dt} = kQ with Q(0)=50Q(0) = 50 and Q(4)=200Q(4) = 200. Find Q(6)Q(6). Give an exact answer.
Q4·Straightforward
A substance decays according to M=M0ektM = M_0 e^{kt}. The initial mass is 400 g and the half-life is 6 hours (i.e., M(6)=200M(6) = 200). Find the value of kk to 4 decimal places.
Q5·Moderate
A metal rod is heated to 120°C and placed in a room at 20°C. It cools according to Newton's law of cooling: T(t)=20+100 e−ktT(t) = 20 + 100\,e^{-kt}. After 10 minutes the temperature is 70°C. Find the temperature after 20 minutes (in °C).
Q6·Moderate
A bacterial culture grows from 500 to 2000 cells in 4 hours. Using N=N0ektN = N_0 e^{kt}, find the number of cells after 7 hours. Round to the nearest whole number.
Q7·Moderate
The half-life of carbon-14 is 5730 years. An ancient wooden artefact contains 30% of its original carbon-14. Find the age of the artefact to the nearest year. Use k=−ln⁡25730k = -\dfrac{\ln 2}{5730}.
Q8·Moderate
A cup of tea cools from 90°C to 50°C in 8 minutes in a room at 20°C. Using Newton's law of cooling in the form T(t)=20+70 e−ktT(t) = 20 + 70\,e^{-kt}, find kk to 4 decimal places.
Q9·Moderate
Show that the function N=N0ektN = N_0 e^{kt} satisfies the differential equation dNdt=kN\dfrac{dN}{dt} = kN and the initial condition N(0)=N0N(0) = N_0.

✎ Work this one through on paper — proofs are self-assessed.

Q10·Moderate
A population satisfies dPdt=k(P−500)\dfrac{dP}{dt} = k(P - 500). Initially P=2000P = 2000 and after 5 years P=3200P = 3200. Find PP to the nearest whole number when t=10t = 10.
Q11·Challenging
Carbon-14 dating. A sample of wood from an archaeological site contains 60% of the carbon-14 that would be present in living wood. The half-life of carbon-14 is 5730 years. Find the age of the sample to the nearest year.
Q12·Challenging
Two populations AA and BB grow according to PA(t)=1000 e0.1tP_A(t) = 1000\,e^{0.1t} and PB(t)=2000 e0.05tP_B(t) = 2000\,e^{0.05t}, where tt is in years. After how many full years will population AA first exceed population BB?
Q13·Challenging
A patient is given a drug that is eliminated from the bloodstream at a rate proportional to the current concentration. The initial concentration is 80 mg/L and after 4 hours it is 50 mg/L. Find the concentration (in mg/L, to 2 decimal places) when t=10t = 10 hours.