AHL 2.10—Scaling large numbers, log-log graphs Questionbank

A geologist models the decay of seismic wave amplitude $A$ with distance $d$ (in km) from an earthquake epicenter, using $A=k d^{n} e^{m d}$, where $k, n, m$ are constants. Data is plotted as $\ln A$ against $\ln d$, but due to the exponential term, the graph is not perfectly linear. However, for small $d$, the term $e^{m d} \approx 1$, and the plot approximates a straight line through $(0,5)$ and $(1,2)$.

Graph of $\ln A$ against $\ln d$

A chemist studies the rate $R$ of a reaction as a function of concentrations $A$ and $B$ of two reactants, modeled by $R=k A^{x} B^{y}$. The following data is collected:

The brightness $B$ of a star, measured in arbitrary units, is believed to be related to its distance $d$ (in light-years) from Earth by the formula $B=\frac{k}{d^{n}}$, where $k$ and $n$ are positive constants. An astronomer plots $\log _{10} B$ against $\log _{10} d$ and obtains the following graph:

Graph of $\log _{10} B$ against $\log _{10} d$

A physicist models the decay of a radioactive substance, where the mass $M$ (in grams) remaining after time $t$ (in days) is given by $M=k e^{m t}$, with $k, m$ constants. The physicist plots $\ln M$ against $t$ and finds a straight line passing through (2, 4.605) and (5, 3.912).

A scientist models the electrical resistance $R$ (in ohms) of a conductor as a function of its length $L$ (in meters), using $R=k L^{n}$. A plot of $\ln R$ against $\ln L$ is linear through ( 0 , 2.303) and (1, 3.912).

A researcher models the heat transfer rate $Q$ (in $\mathrm{J} / \mathrm{s}$ ) through a material as a function of its thickness $x$ (in cm), using $Q=k x^{n}$. A $\log$-log plot of $\log _{10} Q$ against $\log _{10} x$ is a straight line through (0.301, 2.000) and (0.602, 1.301).

The following data represents the population of a certain species of fish in a lake over a period of 10 years. The population is believed to grow exponentially.

The force, $F$ Newtons, between two magnets a distance $d$ metres apart is modelled by the equation $F = k d^n$. The following measurements were taken:

$\begin{array}{|c|c|} \hline d & F \\ \hline 1 & 0.15 \\ 2 & 0.0375 \\ 3 & 0.0167 \\ 4 & 0.00938 \\ \hline \end{array}$

Consider a dataset that represents the growth of a bacterial culture over time. The number of bacteria $N$ at time $t$ (in hours) is recorded. The data is suspected to follow an exponential growth pattern.