The Minimum Possible Surface Area of A Tetrahedron with A Fixed Face and Volume

Geometry Level 5

Let \(ABC\) be a triangle such that \(BC=13\), \(CA=14\), and \(AB=15\). Furthermore, let \(D\) be a point in space such that the volume of tetrahedron \(ABCD\) is \(28\). The minimum possible surface area of tetrahedron \(ABCD\) can be expressed in the form \(p\sqrt{q}+r\), where \(p\) and \(r\) are positive integers and \(q\) is a squarefree integer. Find \(p+q+r\).

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