A man is walking along a straight path. A searchlight is located at S, which is 40 metres from the nearest point Q on the path. The searchlight can rotate, and is kept focused on the man as he walks. When the rotation angle QSP of the spotlight is radians, the man is at a point P on the path which is x metres from Q as shown on the diagram. Note that QP S is a right triangle.

(a) [2 points] The distance x depends on the angle . In other words x = x(). Find the function x(), and find its derivative dx d = x 0 ().

(b) [1 points] At time t the searchlight is at angle = (t) for some (unknown) function (t), so the searchlight's rate of rotation is d dt = 0 (t) radians per second. Therefore man's position at time t is x((t)). Using your answer to part (a), express the man's walking speed, dx dt , in terms of (t) and 0 (t)

(c) [1 point] What is the value of sec() when x() = 30?

(d) [2 points] Suppose that, at time t0, the man is 30 metres from Q and the searchlight is rotating at 4 125 radians per second. How fast is the man walking at time t0?