7. The time, $T$ seconds, that a pendulum takes to complete one swing is modelled by the formula

$$T = a l ^ { b }$$

where $l$ metres is the length of the pendulum and $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Show that this relationship can be written in the form

$$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{59120894-c480-492b-a304-106ddbadacf0-18_613_926_699_699}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

A student carried out an experiment to find the values of the constants $a$ and $b$.\\
The student recorded the value of $T$ for different values of $l$.\\
Figure 3 shows the linear relationship between $\log _ { 10 } l$ and $\log _ { 10 } T$ for the student's data. The straight line passes through the points $( - 0.7,0 )$ and $( 0.21,0.45 )$

Using this information,
\item find a complete equation for the model in the form

$$T = a l ^ { b }$$

giving the value of $a$ and the value of $b$, each to 3 significant figures.
\item With reference to the model, interpret the value of the constant $a$.\\[0pt]
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[TURN OVER FOR QUESTION 8]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q7 [6]}}