[It is given that the volume of a cylinder of radius $r$ and height $h$ is $\pi {{r}^{2}}h$.] The di

Question

[It is given that the volume of a cylinder of radius $r$ and height $h$ is $\pi {{r}^{2}}h$.] The diagram above shows a cylinder of radius $r$ cm and height $h$ cm inscribed in a spherical container with fixed radius $5$cm. Show that the volume, $V$cm$^{3}$, of the cylinder is given by $V=2\pi {{r}^{2}}\sqrt{25-{{r}^{2}}}$. [2] As $r$ varies, use differentiation to find the exact value of $r$ that gives a maximum $V$ of the cylinder. Hence find the exact maximum value of $V$, showing that this value is a maximum. [6] The diagram shows the graph $y=\mathrm{f}(x)$, which is strictly decreasing on the interval $(-2,1)$. Given that $\mathrm{g}(x)=\frac{1}{2}\sin [2\mathrm{f}(x)]$, where $-2 [4]

Accepted Answer

$\dfrac{500\sqrt{3}\,\pi}{9}$ cm³; one stationary point

2019 TMJC Promo Q9

2019 TMJC Promo Q9

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