To find the volume of this cone, we need to know the height of the cone, and the formula for the volume of a cone.

First, to find the height of the cone, we use the information given in the diagram. The diameter of the entire cone is 10 inches, which means the radius is 5 inches. We are also told the slant of the cone is 13 inches. With these pieces of information, we can use the Pythagorean Theorem to find the measure of the height.

Now that we have solved for the height of the cone, we can use the formula for the volume of a cone to find our solution. The volume of a cone is:

So the correct choice is A.