<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

The table gives the velocity $v ms^{-1}$ of an electric milk float
$t$ seconds after starting from rest.

\begin{tabular} {|c|c|c|c|c|c|c|c|}
\hline

$t$ &amp; 0 &amp; 2 &amp; 4 &amp; 6 &amp; 8 &amp; 10 &amp; 12\\

\hline

$v$ &amp; 0 &amp; 3.0 &amp; 4.7 &amp; 5.8 &amp; 6.6 &amp; 7.1 &amp; 7.3\\

\hline
\end{tabular}

Use Simpson's rule to estimate the distance travelled in 12
seconds.
\medskip

{\bf Answer}

Now distance travelled in time $T$, $S=\ds\int_0^Tv \,dt$

Here $T=12$ and we have 7 ordinates $y_1 \rightarrow y_7$ and from
table the spacing of $x$ values gives $h=2$

$\begin{array} {rcl} S &amp; = &amp; \ds\frac{2}{3}
\times(\underbrace{0+7.3}+4(\underbrace{3.0+5.8+7.1})+2(\underbrace{4.7+6.6}))\\
&amp; &amp; \ \ \ \ \ \ \ \ y_1+y_7 \ \ \ \ \ \ \ \ y_2+y_4+y_6 \ \ \ \ \
\ \ \ \ \ y_3+y_5\\ &amp; = &amp; \un{62.333=62m} \rm{to\ 3s.f.}
\end{array}$
\end{document}
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