6 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]\\
The variables $x$ and $y$ are known to be related by an equation of the form

$$y = k x ^ { n }$$

where $k$ and $n$ are constants.\\
Experimental evidence has provided the following approximate values:

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 4 & 17 & 150 & 300 \\
\hline
$y$ & 1.8 & 5.0 & 30 & 50 \\
\hline
\end{tabular}
\end{center}

(a) Complete the table in Figure 1, showing values of $X$ and $Y$, where

$$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$

Give each value to two decimal places.\\
(b) Show that if $y = k x ^ { n }$, then $X$ and $Y$ must satisfy an equation of the form

$$Y = a X + b$$

(c) Draw on Figure 2 a linear graph relating $X$ and $Y$.\\
(d) Find an estimate for the value of $n$.

\hfill \mbox{\textit{AQA FP1  Q6}}