So point C is sliding only along the line AB.

The warning then will sound if point C is at 0.5 units to the right of point A.

Let's call point C' the position of point C when it is 0.5 units to the right of A.

And point D' the position of point D at that instant.

If we can get the x-coordinate of C', then that is also the x-coordinate of D'. Then we can get the y-coordinate of D' by using the circle equation.

Draw a radius to A.

By Pythagorean theorem,

4^2 = (3 -2)^2 +(AC)^2

AC = sqrt(16 -1) = 3.87298

So the coordinates of A are ((1 -3.87298),3) = (-2.87298,3)

Then the position of C' is ((-2.87298 +0.5),3) = (-2.37298,3)

Hence the x-coordinate of D' is -2.37298 also.

In the circle,

(x -1)^2 +(y -2)^2 = 16,

(-2.37298 -1)^2 +(y -2)^2 = 16

(y -2)^2 = 16 -(-3.37298)^2

y -2 = sqrt(16 -11.37699)

y = 2.15012 +2

y = 4.15012

Hence, D' is (-2.37298,4.15012)

Therefore, whenever D is between, and including, points (-2,37298,4.15012) and (-2.87298,3) along the circumference of the circle, the alarm rings. ----answer.