Word/applied problems (growth, decay, cooling, mixing, circuits)

MATApplied Problems

Applied problems model a real quantity whose rate of change is proportional to itself or to a difference, leading to first-order separable or linear equations. The core skill is translating the words into $\dfrac{dy}{dt}=ky$ (growth/decay), Newton's law of cooling, a mixing balance, or an RL-circuit equation, then solving and fitting the constants to the given data.

ISC examines these for both the modelling step and clean exponential answers, often expressed using $\ln$ .

Exponential growth / decay

\dfrac{dN}{dt}=kN\ \Rightarrow\ N=N_{0}\,e^{kt}

N_0

is the initial amount,

k>0

for growth and

k<0

for decay; find

k

from a given doubling/percentage data point.

Newton's law of cooling

\dfrac{d\theta}{dt}=-k(\theta-\theta_{s})\ \Rightarrow\ \theta-\theta_{s}=(\theta_{0}-\theta_{s})\,e^{-kt}

\theta

is body temperature,

\theta_s

the surrounding temperature,

\theta_0

the initial temperature,

k>0

Mixing (tank) balance, equal flow rates

\dfrac{dS}{dt}=(\text{rate in})-(\text{rate out})=c\,r-\dfrac{r}{V}\,S

S(t)

is salt amount,

c

the inflow concentration,

r

the flow rate,

V

the (constant) tank volume; this is linear in

S

RL circuit current

L\dfrac{di}{dt}+Ri=E\ \Rightarrow\ i=\dfrac{E}{R}\Big(1-e^{-Rt/L}\Big)

L

inductance,

R

resistance,

E

constant EMF, with

i=0

t=0

; steady-state current is

\dfrac{E}{R}

Half-life from a decay rate

t_{1/2}=\dfrac{\ln 2}{|k|}

|k|

is the decay constant; time for the amount to fall to half its value.

Step 1: name the quantity and write its rate equation from the words; 'rate proportional to amount' means $\dfrac{dN}{dt}=kN$ .
Growth/decay solves to $N=N_0 e^{kt}$ ; use one data point (e.g. doubling time, percent lost) to solve for $k$ in terms of $\ln$ .
Doubling: $2N_0=N_0 e^{kT}\Rightarrow k=\dfrac{\ln 2}{T}$ ; tripling time is then $t=\dfrac{\ln 3}{k}=T\dfrac{\ln 3}{\ln 2}$ .
For cooling, work with the difference $\theta-\theta_s$ , which decays exponentially with the same $k$ across each interval.
Mixing problems give a linear ODE $\dfrac{dS}{dt}+\dfrac{r}{V}S=cr$ ; solve by integrating factor and find the long-run value as $t\to\infty$ .
In a tank with equal in/out rates the volume $V$ stays constant, so the outflow concentration is $\dfrac{S}{V}$ .
The RL solution rises from $0$ toward the steady state $\dfrac{E}{R}$ ; the time constant is $\dfrac{L}{R}$ .
Continuously compounded growth uses $\dfrac{dA}{dt}=rA\Rightarrow A=A_0 e^{rt}$ , and doubling time is $\dfrac{\ln 2}{r}$ .

Where the marks go

Sign error in cooling/decay: using $+k$ instead of $-k$ so the quantity grows instead of decaying.
Trying to find a numerical $k$ when the data only fixes a ratio — keep the answer in terms of $\ln 2,\ln 3$ as the question asks.
In mixing problems, forgetting that the outflow concentration is $\dfrac{S}{V}$ (not the inflow concentration), or letting $V$ vary when inflow and outflow rates are equal.
Skipping the initial condition: not using $i=0$ at $t=0$ (or $N=N_0$ at $t=0$ ) leaves an undetermined constant.

How the board asks it

Applicationgrowth/decay model $\dfrac{dN}{dt}=kN$
A bacterial culture grows at a rate proportional to its size and doubles in $4$ hours. If the initial population is $N_0$ , find the population after $10$ hours and the time taken to triple, leaving your answer in terms of $\ln 2$ and $\ln 3$ .
Numericalhalf-life from a decay rate
A radioactive substance decays so that $20\%$ is lost in $5$ years. Form and solve the differential equation $\dfrac{dN}{dt}=-kN$ , and hence calculate the half-life of the substance.
Derive / proveNewton's law of cooling on the difference $\theta-\theta_s$
A body at $90^\circ C$ is placed in a room at $20^\circ C$ and cools to $60^\circ C$ in $10$ minutes. Using Newton's law of cooling, obtain an expression for its temperature $\theta$ at time $t$ and find the temperature after a further $10$ minutes.
Applicationmixing balance $\dfrac{dS}{dt}+\dfrac{r}{V}S=cr$
A tank holds $100$ litres of pure water. Brine containing $2$ g of salt per litre flows in at $5$ litres/min and the well-stirred mixture flows out at the same rate. Set up and solve the differential equation for the amount of salt $S(t)$ , and find the salt content as $t\to\infty$ .
Derive / proveRL circuit current toward steady state $\dfrac{E}{R}$
In an $L$ - $R$ circuit the current $i$ satisfies $L\dfrac{di}{dt}+Ri=E$ with $i=0$ at $t=0$ . Solve for $i(t)$ and find the time, in terms of $\dfrac{L}{R}$ , at which the current reaches half its steady-state value.

Practise this topic

PreviousLinear differential equations (integrating factor)

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.