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) The distance x depends on the angle . In other words x = x(). What is the function x(), and find its derivative dx/d = x 0 ().

b) 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 part (a), express the man's walking speed, dx/dt , in terms of (t) and 0 (t).

c) What is the value of sec() when x() = 30?

d) 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?