C3 June 2013 Q8

EdexcelOld spec14 marksModellingTrigonometry

8.

Figure 2: a road of width 7 m; Kate walks from A on one edge at speed V m/s at angle theta to the edge, reaching B on the other edge; John runs at 3 m/s along the road, starting 24 m behind A
Figure 2

Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s−1.
Kate is 24 m ahead of John when she starts to cross the road from the fixed point \(A\).
John passes her as she reaches the other side of the road at a variable point \(B\), as shown in Figure 2.
Kate’s speed is \(V\) m s−1 and she moves in a straight line, which makes an angle \(\theta\), \(0<\theta<150^\circ\), with the edge of the road, as shown in Figure 2.

You may assume that \(V\) is given by the formula\[V=\frac{21}{24\sin\theta+7\cos\theta},\qquad 0<\theta<150^\circ\]

(a) Express \(24\sin\theta+7\cos\theta\) in the form \(R\cos(\theta-\alpha)\), where \(R\) and \(\alpha\) are constants and where \(R>0\) and \(0<\alpha<90^\circ\), giving the value of \(\alpha\) to 2 decimal places. (3)

Given that \(\theta\) varies,

(b) find the minimum value of \(V\). (2)

Given that Kate’s speed has the value found in part (b),

(c) find the distance \(AB\). (3)

Given instead that Kate’s speed is 1.68 m s−1,

(d) find the two possible values of the angle \(\theta\), given that \(0<\theta<150^\circ\). (6)
‹ Previous56 of 144 in ModellingNext ›