What is the locus of points that a dog may reach when it is tied at the center of a 30-foot fence with an 8-foot leash?

The locus of points would be a semicircle with a radius of 8 ft. The locus of points would then be [x] = [-8,8] and [y] = [8,0].

To demontrate this we can write the equation of a semicircle. We can start by writing the equation of a circle:

x2+y2 = r2

For the purposes of this question we will label the fence as the x-axis, and the direction away from the fence as the positive y-axis. In a semi-cricle, the possible values of y are [r,0] so to show this in the equation we should rearrange for y:

y2=r2-x2 --> y = √(r2-x2) --> y=√(64-x2)

As the square root of a number is always positive, we have successfully demonstrated that the locus of y points is [8,0].