In the diagram, point \(Q\) is 400 m vertically above point \(A\). Initially, a balloon is at point \(Q\) and moves horizontally in a straight line at a constant speed of 10 m per second away from an observer stationed at point \(A\). At time \(t\) seconds after the start, the balloon is at point \(P\), where \(QP\) is \(x\) m and the angle of elevation of the balloon from the observer at \(A\) is \(\beta\) radians. Find exactly the rate of change of \(\beta\) at the instant when \(x\) is 100 m.[4]
A gardener designs a flower bed \(ABC\) in the shape of an isosceles triangle inscribed in a circular plot of land with radius 6 metres, as shown in the diagram. The vertices of the triangle lie on the circumference of the circle and \(AB=AC\). The angle \(BAC\) is \(\theta\), where \(\theta\) is acute. Express the area of the flower bed \(ABC\) in terms of \(\theta\).[2]
As \(\theta\) varies, use differentiation to find the exact maximum possible area of the flower bed, and show that it is a maximum.[4]
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Answer:(a) \(-\dfrac{2}{85}\text{ rad s}^{-1}\) (b) \(A=36(1+\cos\theta)\sin\theta\text{ m}^2\), maximum \(27\sqrt3\text{ m}^2\)