## Exam-Style Question on Mensuration## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 570. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.

An Asian water buffalo is tethered in a rectangular field by a rope of length \(r\) metres. One end of the rope is securely tied to point \(P\) as shown in this diagram (not drawn to scale).

Points \(Q\) and \(R\) on the fence enclosing the field, are directly opposite each other and are the furthest points the buffalo can reach on the edge of the field. \(PQ =PR = r\) and the angle \(Q\hat{P}R = \theta\) radians. The length of arc QR shown is 38m.

(a) Write down an expression for \(r\) in terms of \(\theta\).

(b) Show that the area of the field that the buffalo can reach is \( \frac{1444}{\theta^2} ( \frac{\theta}{2} + \sin \theta) \)

(c) The area of the field that the buffalo can reach is 900m^{2}. Find the value of \(\theta\).

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