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?