Let $C>0$. A dimensionless index $R$ is defined by

The radius of a circular garden is measured as $8.4$ m, correct to the nearest $0.1$ m. A landscape designer calculates its area using $A=\pi(8.4)^2$.

A rectangular solar panel has a measured length of $1.80$ m and a measured width of $0.95$ m. Both measurements are correct to the nearest $0.01$ m. The area calculated from the measured dimensions is $1.71\ \text{m}^2$.

The pH of a solution with hydrogen ion concentration $H$ is given by

Solution A has concentration $H_A=3.20\times10^{-5}$ and solution B has concentration $H_B=8.00\times10^{-7}$.

The mass $M$, in grams, of an artificial model is related to its length $L$, in centimetres, by

where $k$ is a positive constant. A model of length $8$ cm has mass $96$ g.

The magnitude $M$ of an earthquake is modelled by

where $A$ is its measured amplitude and $A_0$ is a reference amplitude. Earthquake P has amplitude $2.50\times10^4A_0$, and earthquake Q has amplitude $4.00\times10^6A_0$.

Consider the equation

A space probe collects $3.75\times10^{22}$ dust particles. The mean mass of one particle is $6.40\times10^{-17}\ \text{kg}$. The probe also collects $8.60\times10^5\ \text{kg}$ of larger fragments.

The response $D$ of a light sensor is modelled by
$D=\log\left(\frac{I}{I_0}\right)$
where $I$ is the light intensity and $I_0=2.50\times10^{-6}\ \text{W m}^{-2}$.

An uncompressed digital map contains $2^{30}$ bytes. A compression process multiplies its size by $2^{-8}$, after which indexing data multiplies the resulting size by $2^3$.

A liquid medicine has a measured volume of $12.4\ \text{mL}$, correct to the nearest $0.1\ \text{mL}$. Its measured concentration is $2.35\ \text{mg mL}^{-1}$, correct to the nearest $0.01\ \text{mg mL}^{-1}$. A pharmacist reports that the dose contains $29.1\ \text{mg}$ of medicine.

A relief organization orders three types of supply crate: A, B and C. Let $x$, $y$ and $z$ be the numbers of each type ordered. There are $120$ crates in total. The numbers of food units and medical units give the equations
$2x+y+3z=220$
and
$x+4y+2z=300$
Crates A, B and C cost 18, 24 and 31 euros respectively.

During a $4.5$-hour calibration test, the signed positioning error of a robotic arm is modelled by
$p(t)=t^4-10t^3+35t^2-50t+24$
where $t$ is measured in hours and $0\le t\le4.5$. A positive value of $p(t)$ means that the arm is above its target position.

A cylindrical storage vessel has measured diameter $24.6\ \text{cm}$, correct to the nearest $0.1\ \text{cm}$, and measured height $38\ \text{cm}$, correct to the nearest centimetre. Its volume is calculated using
$V=\pi\left(\frac d2\right)^2h$

Three currents $I_1$, $I_2$ and $I_3$, measured in amperes, satisfy Kirchhoff's equations
$I_1-I_2-I_3=0$
$2I_1+4I_2=18$
$2I_1+5I_3=25$

The signal loss $L$, in decibels, across a component is modelled by
$L=20\log\left(\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}\right)$
where $V_{\mathrm{in}}$ and $V_{\mathrm{out}}$ are positive voltages.

The daily energy requirement $E$, in kilojoules, of an experimental organism is modelled by
$E=km^{\frac34}$
where $m>0$ is its mass in grams and $k$ is a positive constant. An organism of mass $16\ \text{g}$ requires $240\ \text{kJ}$.

The apparent magnitude $m$ of a star is modelled by
$m=-2.5\log\left(\frac{F}{F_0}\right)$
where $F$ is the observed flux and $F_0$ is a reference flux. Star A has flux $4.00\times10^{-5}F_0$ and star B has flux $2.50\times10^{-7}F_0$.

An observatory records $360$ images. Each image contains $2.40\times10^8$ pixels and each pixel requires $14$ bits. The recorded data are compressed to $17.5\%$ of their original size. The compressed data are transmitted through a link advertised at $6.25\times10^7$ bits per second, but the link operates at $82.0\%$ of this rate.

A filtration process begins with a particle concentration of $6.40\times10^5$ particles per millilitre. During each filtration cycle, the concentration is multiplied by $8^{-\frac12}$. After $n$ cycles, the concentration is $C_n$.

A rectangular water tank has measured length $1.84$ m and measured width $0.96$ m, each correct to the nearest $0.01$ m. Its measured depth is $0.425$ m, correct to the nearest $0.001$ m. The volume calculated from the reported measurements is $0.75072\ \text{m}^3$.

A recycling plant produces a $100$ kg batch by mixing three materials, $A$, $B$ and $C$. Their metal contents are $80\%$, $50\%$ and $20\%$, respectively. Their processing costs are $7$, $5$ and $3$ euros per kilogram, respectively. The batch contains $51.5$ kg of metal and costs $510$ euros to process. Let $x$, $y$ and $z$ be the masses, in kilograms, of $A$, $B$ and $C$.

The signed calibration error of a sensor is modelled by
$f(t)=t^3-7t^2+14t-8$
where $t$ is the time in hours and $0\le t\le5$. A negative value indicates that the sensor reads below the reference value.

The apparent brightness index $m$ of an astronomical object is defined by
$m=-2.5\log\left(\frac{F}{F_0}\right)$
where $F$ is the observed flux and $F_0$ is a reference flux. Two unresolved stars have fluxes $F_A=3.20\times10^{-8}F_0$ and $F_B=7.50\times10^{-9}F_0$.

The strength index $S$ of a composite material is modelled by
$S=k\rho^{\frac32}d^{-\frac12}$
where $\rho>0$ is its relative density, $d>0$ is its grain-size index and $k>0$ is a constant. A sample with $\rho=4$, $d=9$ and $S=80$ is used for calibration.

A distribution centre sends parcels along three routes, $X$, $Y$ and $Z$. During one hour, a total of $120$ parcels are sent. A workload index gives the equations
$3x+2y+z=230$
and
$x+4y+2z=310$
where $x$, $y$ and $z$ are the numbers of parcels sent along the routes.

The wingbeat frequency $f$ of a mechanical flying model is represented by
$f=km^{-\frac13}L^{-\frac12}$
where $m>0$ is a normalized mass, $L>0$ is a normalized wing length and $k>0$ is a constant. A model with $m=8$, $L=9$ and $f=10$ is used for calibration.

A dimensionless security parameter $x$ satisfies
$\ln(x-2)+2\ln(x+1)=\ln108$