<p>The diagram below (not drawn to scale) shows the vertical cross section of a cylindrical drum with centre $O$ and radius $5$cm. The drum is fixed at a point $B$ on the horizontal ground and a rectangular metal sheet leans on it.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1759739938187-bv32e873apb.png" width="400" class="aligncenter" /></figure>
<p>Initially, the metal sheet touches the horizontal ground at point $P$ such that $PB=50$cm and $\angle OPB=\theta $ radians. The metal sheet is moved in the direction $PB$ so that $P$ is $h$ cm closer to $B$ and $\angle OPB$ is increased by $\alpha $ radians, where $\alpha $ is small.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using a suitable compound angle formula in MF26, show that</div><div class="list-item-paragraph" style="text-align: center">$h\approx \frac{505\alpha }{1+10\alpha }$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence express $h$ as a series in $\alpha $, up to and including the term in ${{\alpha }^{3}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $P$ is moving towards $B$ at a rate of $0.1$cms-1, find the rate of change of $\alpha $ when $h=1$.</div><p style="text-align: right;">[4]</p></li></ol>

The diagram below (not drawn to scale) shows the vertical cross section of a cylindrical drum with centre $O$ and radius $5$cm. The drum is fixed at a point $B$ on the horizontal ground and a rectangular metal sheet leans on it.

Initially, the metal sheet touches the horizontal ground at point $P$ such that $PB=50$cm and $\angle OPB=\theta $ radians. The metal sheet is moved in the direction $PB$ so that $P$ is $h$ cm closer to $B$ and $\angle OPB$ is increased by $\alpha $ radians, where $\alpha $ is small.

Using a suitable compound angle formula in MF26, show that

$h\approx \frac{505\alpha }{1+10\alpha }$.

Hence express $h$ as a series in $\alpha $, up to and including the term in ${{\alpha }^{3}}$.

Given that $P$ is moving towards $B$ at a rate of $0.1$cms-1, find the rate of change of $\alpha $ when $h=1$.

2020 TJC P1 Q8

2020 TJC P1 Q8

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