A Cone and a Sphere!

Geometry Level 4

In the diagram above, the sphere has a diameter of \(10 \text{ cm}\). Also, the right circular cone has a height of 10 cm, and it's base has a diameter of \(10 \text{ cm}\). The sphere and the cone stand on a horizontal surface. If a horizontal plane cuts both the sphere and the cone, the cross-sections will both be circles as shown.

Find the height of the horizontal plane (from the bottom) (in \(\text{cm}\)) that gives circular cross-sections of the sphere and cone of equal area. Note that the height should be less than \(10 \text{ cm}\).

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Bonus: A sphere of diameter \(d\) and a right circular cone with a base of diameter \(d\) stand on a horizontal surface. In this case, the height of the cone is equal to the radius of the sphere. Show that, for any horizontal plane that cuts both the cone and the sphere, the sum of the areas of the circular cross-sections is always the same.


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