To find the radius of the cone, we can use the information given in the problem.
Let's consider the cross-section of the frustum. In the cross-section, we have a large circle (base of the frustum), a smaller circle (top of the frustum), and a triangle connecting the two bases (side of the frustum).
Since the sphere is tangent to both bases and the side of the frustum, it means that the radius of the sphere is equal to the radius of both the large and small circles.
Given that the radius of the sphere is 3, this means that the radius of both the large and small circles is also 3.
Now, let's focus on the triangle connecting the bases. The slant height of the frustum is given as 10. The slant height of a cone is the hypotenuse of a right triangle, and the altitude of the cone (perpendicular distance between the bases) is one of the legs.
In our case, the slant height (10) is the hypotenuse, and the altitude is the other leg. We can use the Pythagorean theorem to find the altitude.
Let's call the altitude h. Using the Pythagorean theorem, we have:
h^2 + (3-3)^2 = 10^2
Simplifying, we get:
h^2 = 100
Taking the square root of both sides, we get:
h = 10
Now that we have the altitude (h) and the radii (r), we can find the volume of the frustum.
The formula for the volume of a frustum is given as:
V = (1/3) * pi * h * (R^2 + r^2 + R*r)
where R is the radius of the large circle and r is the radius of the small circle.
In our case, R = 3 (radius of the large circle) and r = 3 (radius of the small circle).
Plugging in the values, we get:
V = (1/3) * pi * 10 * (3^2 + 3^2 + 3*3)
V = (1/3) * pi * 10 * (9 + 9 + 9)
V = (1/3) * pi * 10 * 27
V = pi * 90
So, the volume of the frustum is 90pi cubic units.