??? - 2013 JUNE Q8
8.
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 , 0 < θ < 150, with the edge of the road, as shown in Figure 2.
You may assume that V is given by the formula
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 , 0 < θ < 150, with the edge of the road, as shown in Figure 2.
You may assume that V is given by the formula
(a) Express 24sinθ + 7cosθ in the form Rcos(θ–a), where R and a are constants and where R > 0 and 0 < a < 90, giving the value of to 2 decimal places. (3)
Given that θ 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 that θ 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 θ, given that 0 < θ < 150. (6)
(d) find the two possible values of the angle θ, given that 0 < θ < 150. (6)