Each side of an equilateral triangle increases from an initial length of $9$ cm at a steady rate o

Question

Each side of an equilateral triangle increases from an initial length of $9$ cm at a steady rate of $0.1\mathrm{ cm }{{\mathrm{s}}^{-1}}$. Find the rate of increase of the area of the equilateral triangle after $30$seconds, giving your answer in exact form.[4] The diagram above shows a line with negative gradient that passes through the point $P\left( 2,3 \right)$ and meets the $x$-axis and $y$-axis at $A$ and $B$ respectively. It is also given that $\angle OAB=\theta $radians. Show that the length$AB=2\sec \theta +3\mathrm{cosec}\theta $.[1] Find, using differentiation, the value of $\theta $ which gives a minimum length of $AB$.[5]

Accepted Answer

(a) $\frac{3\sqrt3}{5}\text{ cm}^2\text{/s}$ (b)(i) $AB=2\sec\theta+3\cosec\theta$ (b)(ii) $\theta=\tan^{-1}\left(\sqrt[3]{\frac32}\right)\approx0.853\text{ rad}$

2013 MJC P2 Q3

2013 MJC P2 Q3

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