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Exam-Style Question on Optimisation

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Question id: 657. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

A circle with equation \(x^2+y^2=25 \) has centre \((0,0)\) and radius 5.

A trapezium, ABCD, is inscribed in the circle with its vertices at \( A(x,y), B(4,-3), C(-4,-3) \text{ and } D(-x,y)\), where A and D are variable points in the first and second quadrants respectively. This is shown in the following sketch.

Trapezium in Circle

(a) For point A, show that \(y = \sqrt{25-x^2} \).

(b) Hence, find an expression for T, the area of trapezium ABCD, in terms of x.

(c) Show that \( \dfrac{dT}{dx} = 3 - \dfrac{2x^2 +4x -25}{\sqrt{25-x^2}} \).

(d) Hence or otherwise, find the value of \(x\) such that T is a maximum.

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