The mass, $M$ grams, of leaves on a plant $t$ days after an experiment begins is modelled by

$M(t)=at^2+bt+c$

The measured masses are $12$ grams when $t=0$, $28$ grams when $t=4$, and $22$ grams when $t=10$.

The sound level, $S$ decibels, produced by a machine operating at power $x$ kilowatts is modelled by

$S(x)=a+b\ln x$

where $x>0$. The sound level is $18$ decibels when $x=2$ and $42$ decibels when $x=10$.

The population, $N$, of insects in a greenhouse is modelled by

$N(t)=\frac{1200}{1+Ce^{-0.35t}}$

where $t$ is the number of weeks after observations begin. Initially there are $150$ insects.

The cost, $C(x)$ euros, of delivering a package a distance of $x$ kilometres is modelled by

The initial delivery charge is €$5$, and the model is continuous at $x=10$.

The number, $y$ thousands, of daily visits to a website was recorded over time. A best-fit straight line for a plot of $\ln y$ against time $t$ days is

$\ln y=1.47+0.082t$

Data for positive variables $x$ and $y$ were analysed using three transformations. The following correlation coefficients were obtained:

• a plot of $y$ against $x$: $r=0.941$;
• a plot of $\ln y$ against $x$: $r=0.998$;
• a plot of $\ln y$ against $\ln x$: $r=0.972$.

For the second plot, the best-fit line is

$\ln y=2.20+0.315x$

The observed values of $x$ were between $1$ and $6$.

A force $F$ newtons and speed $v$ metres per second are believed to satisfy a power model

$F=Kv^p$

A best-fit line for a plot of $\log_{10}F$ against $\log_{10}v$ is

$\log_{10}F=-0.602+1.75\log_{10}v$

An electric bicycle company charges according to the duration, $t$ minutes, for which a bicycle is hired. The cost, $C(t)$ dollars, is modelled by

The cost model has no jump at $t=30$. A competing company charges $D(t)=9+0.07t$ dollars for $0\leq t\leq120$.

The value, $V(t)$ dollars, of a specialized camera $t$ years after purchase is modelled by

$V(t)=ke^{-rt}+120$

where $k>0$, $r>0$ and $t\geq0$. The camera initially costs $1320 and has a value of $780 after three years.

On a particular wet road, the stopping distance, $s$ metres, of a car varies directly with the square of its speed, $v$ kilometres per hour. The model is

$s=kv^2$

At $50\ \text{km h}^{-1}$, the stopping distance is $28$ m.

The cross-section of a decorative entrance is modelled by the quadratic function

$h(x)=ax^2+bx+c$

where $h$ is the height in metres and $x$ is the horizontal distance in metres from the left-hand end. The curve passes through $(0,1.2)$, $(4,5.6)$ and $(10,1.2)$.

An open rectangular box is made from a $30$ cm by $20$ cm sheet of card by cutting squares of side $x$ cm from each corner and folding up the sides. The volume is modelled by

$V(x)=x(30-2x)(20-2x)$

where $0<x<10$.

The temperature, $T(t)$ in degrees Celsius, inside a greenhouse is modelled over a repeating daily cycle. The variable $t$ is the number of hours after 15:00. The model is

$T(t)=a\cos(bt^\circ)+d$

The maximum temperature is $32^\circ\text{C}$ at $t=0$, the minimum temperature is $14^\circ\text{C}$, and the period is $24$ hours. Assume that $b>0$.

The yield, $Y$ tonnes per hectare, of a crop is modelled by

$Y(x)=ax^2+bx+c$

where $x$ is the mass of fertilizer applied in hundreds of kilograms per hectare. Measurements give $Y(0)=2.4$, $Y(4)=5.8$ and $Y(10)=4.0$.

The surface area, $A$ square metres, covered by a water plant was recorded every two days from day $0$ to day $6$. The results are shown in the table.

The count rate, $R(t)$ counts per minute, from a sample is modelled by

$R(t)=12+Ae^{-kt}$

where $t$ is measured in hours. The constant value $12$ represents background radiation. Initially the count rate is $212$, and after six hours it is $82$.

The number of bird species, $S$, observed in a protected region of area $A$ square kilometres is modelled by

$S(A)=a+b\ln A,\qquad A>0$

A region of area $5\ \text{km}^2$ contains $42$ species, while a region of area $80\ \text{km}^2$ contains $70$ species.

The charge, $Q(t)$ percent, of a battery during a rapid-charging process is modelled by

where $t$ is measured in minutes. The model is continuous at $t=8$.

A streaming company models the cumulative number of subscribers, $N(t)$, by
$N(t)=\frac{80000}{1+Ce^{-kt}}$
where $t$ is the number of months after launch and $C,k>0$. At launch there are $4000$ subscribers. After six months there are $30000$ subscribers.

The mass, $M$ grams, of a biodegradable coating remaining on an object was measured at different times. A linear regression of $\ln M$ on time $t$, measured in days, gives
$\ln M=4.35-0.165t$
The recorded times lie between $0$ and $15$ days.

Engineers investigate how the heat-loss rate, $H$ kilowatts, from a building depends on wind speed, $v$ metres per second. The data were collected for $2\leq v\leq9$. A regression of $\ln H$ on $\ln v$ gives
$\ln H=-1.10+1.60\ln v$

A factory models the daily cost, $C(x)$ dollars, of producing $x$ units by
$C(x)=\begin{cases}120+8x,&0\leq x<20,\\ax+b,&20\leq x<50,\\460+4(x-50),&x\geq50.\end{cases}$
The cost model is continuous at both production thresholds.

The yield, $Y$ tonnes per hectare, of a crop is modelled in terms of fertilizer application, $x$ kilograms per hectare, by
$Y(x)=a+b\ln x,\qquad x>0$
Field trials give $Y(20)=4.5$ and $Y(80)=7.3$.

Radiation detected near a sealed storage unit is modelled by
$R(t)=B+Ae^{-kt}$
where $R$ is measured in counts per minute and $t$ is measured in hours. The background radiation level is $12$ counts per minute. Initially, $R=92$, and after $5$ hours, $R=42$.

The tensile strength, $S$ megapascals, of a polymer is measured after $n$ thousand loading cycles. A regression of $\ln S$ on $n$ gives
$\ln S=6.12-0.184n$
The data were collected for $0\leq n\leq5$.

Hydrologists model river discharge, $Q$ cubic metres per second, in terms of mean river depth, $d$ metres. For measurements with $1.5\leq d\leq8$, a log-log regression gives
$\ln Q=-2.30+2.40\ln d$

The number of daylight hours, $D(t)$, at a location is modelled sinusoidally, where $t$ is the day number of a $365$-day year. The maximum daylight is $15.4$ hours on day $172$, and the minimum daylight is $9.0$ hours.

The vertical position, $H(t)$ metres, of a moving inspection platform is modelled by

$H(t)=a\sin(b(t-c))+d$

where $t$ is measured in seconds and angles are measured in radians. The maximum height is $7.2$ m, the minimum height is $2.8$ m, and the platform crosses its principal axis moving upwards at $t=1.5$ and at the next such crossing, $t=9.5$. Take $b>0$ and choose $0\le c<8\ \text{s}$.