To decorate a children's party table, a chef will use a spherical melon with a diameter measuring 10 cm, which will serve as a support for inserting various candies. He will remove a spherical cap from the melon, and, to ensure the stability of this support, making it difficult for the melon to roll on the table, the chef will make the cut so that the radius $r$ of the circular section of the cut is at least 3 cm. On the other hand, the chef will want to have the largest possible area of the region where the candies will be attached. To achieve all his objectives, the chef should cut the melon cap at a height $h$, in centimeters, equal to (A) $5 - \frac{\sqrt{91}}{2}$ (B) $10 - \sqrt{91}$ (C) 1 (D) 4 (E) 5
To decorate a children's party table, a chef will use a spherical melon with a diameter measuring 10 cm, which will serve as a support for inserting various candies. He will remove a spherical cap from the melon, and, to ensure the stability of this support, making it difficult for the melon to roll on the table, the chef will make the cut so that the radius $r$ of the circular section of the cut is at least 3 cm. On the other hand, the chef will want to have the largest possible area of the region where the candies will be attached.
To achieve all his objectives, the chef should cut the melon cap at a height $h$, in centimeters, equal to\\
(A) $5 - \frac{\sqrt{91}}{2}$\\
(B) $10 - \sqrt{91}$\\
(C) 1\\
(D) 4\\
(E) 5