Find an equation of the largest sphere with center (3,8,1)(3,8,1) that is contained completely in the first octant.

We know that

The largest sphere contained in the first octant has a radius equal to the smallest distance from the center to the coordinates planes
so
Since the center is (3, 8, 1), the plane is 3 units to the right of the yz-plane, 8 units from the xz plane, and 1 units over the xy-plane. 

The radius must be up to 1 units, because if it was greater than 1 units, it would go beyond the first octant.

therefore

the answer is
the equation of a sphere is
(x-3)²+(y-8)²+(z-1)²=1