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.
- Show that this relationship can be written in the form
$$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59120894-c480-492b-a304-106ddbadacf0-18_613_926_699_699}
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\caption{Figure 3}
\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, - 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.
- With reference to the model, interpret the value of the constant \(a\).
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[TURN OVER FOR QUESTION 8]