Answer: Let's label the variables to solve the problem:
Let "r" be the radius of the sphere.
Let "h" be the height of the cone (given as 8 cm).
Let "θ" be the vertical angle of the cone (given as 52 degrees).
Since the highest point of the sphere touches the base of the cone, the center of the sphere will be aligned with the apex of the cone. The line connecting the center of the sphere to the apex of the cone will be perpendicular to the base of the cone, forming a right triangle.
We can use trigonometry to find the radius "r" of the sphere. In this case, the right triangle is formed by the height "h" of the cone, the radius "r" of the sphere, and half of the vertical angle "θ/2".
Using the tangent function:
tan(θ/2) = opposite/adjacent
tan(52/2) = h/r
tan(26) = 8/r
Now, solve for "r":
r = 8 / tan(26)
Using a calculator:
r ≈ 8 / 0.48773 ≈ 16.40 cm
So, the radius of the sphere is approximately 16.40 cm.