7 A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below.
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The points \(P , O\) and \(Q\) lie on a straight line and the angle \(R O S\) is \(\theta\) radians.
A large copy of the logo, with \(P Q = 5\) metres, is to be put on a wall.
7
- Show that the area of the logo, \(A\) square metres, is given by
$$A = \frac { 25 } { 8 } ( \pi - \theta + 2 \sin \theta )$$
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7 - Show that the maximum value of \(A\) occurs when \(\theta = \frac { \pi } { 3 }\)
Fully justify your answer.}
7
- (ii) Find the exact maximum value of \(A\)
7 - Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if \(P Q\) were increased to 10 metres.
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Use the substitution \(u = x ^ { 5 } + 2\) to show that
$$\int _ { 0 } ^ { 1 } \frac { x ^ { 9 } } { \left( x ^ { 5 } + 2 \right) ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 180 }$$