\(X\) and \(Y\) are points on a circle, centre \(O\) with radius \(r\) cm.
\(XZ=(r+2)\text{ cm}\), \(YZ=(2r-5)\text{ cm}\) and angle \(XZY=90^\circ\).
Form a quadratic equation in terms of \(r\) and show that it reduces to \(3r^2-20r+29=0\).[3]
Solve the equation \(3r^2-20r+29=0\).[3]
Explain why one of the solutions in part (a)(ii) cannot be the radius.[1]
The diagram shows a solid hemisphere of radius \(r\) cm.
Given that the solid hemisphere is melted down and recast to form a solid cone with radius \(\dfrac34r\) cm and height \(h\) cm, express \(h\) in terms of \(r\).[3]
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Answer:(a)(i) \(3r^2-20r+29=0\) (shown) (ii) \(r=4.54\) or \(2.13\) (iii) \(r=2.13\) gives a negative length \(YZ\). (b) \(h=\dfrac{32r}9\)