| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 13 |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.3 This is a standard exponential model question requiring logarithmic transformation. Parts (a)-(d) involve routine techniques: writing a linear equation, converting between exponential and logarithmic forms, and solving for specific values. Only part (e) requires any critical thinking about model limitations. The question is slightly easier than average due to its structured, step-by-step nature with clear guidance. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form1.06i Exponential growth/decay: in modelling context |
\includegraphics{figure_2}
A town's population, $P$, is modelled by the equation $P = ab^t$, where $a$ and $b$ are constants and $t$ is the number of years since the population was first recorded.
The line $l$ shown in Figure 2 illustrates the linear relationship between $t$ and $\log_{10} P$ for the population over a period of 100 years.
The line $l$ meets the vertical axis at $(0, 5)$ as shown. The gradient of $l$ is $\frac{1}{200}$.
\begin{enumerate}[label=(\alph*)]
\item Write down an equation for $l$. [2]
\item Find the value of $a$ and the value of $b$. [4]
\item With reference to the model interpret
\begin{enumerate}[label=(\roman*)]
\item the value of the constant $a$,
\item the value of the constant $b$.
\end{enumerate}
[2]
\item Find
\begin{enumerate}[label=(\roman*)]
\item the population predicted by the model when $t = 100$, giving your answer to the nearest hundred thousand,
\item the number of years it takes the population to reach 200000, according to the model.
\end{enumerate}
[3]
\item State two reasons why this may not be a realistic population model. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q14 [13]}}