(Solved): Step 1 The center of the sphere will be the midpoint of the diameter segment. Recall that the midp ...

Step 1 The center of the sphere will be the midpoint of the diameter segment. Recall that the midpoint of a segment between the points \( (u, v, w) \) and \( (p, q, s) \) is given by \[ \left(\frac{u+\mathrm{p} \sim \mathrm{p}}{2}, \frac{\mathrm{v} \sim \mathrm{v}+q}{2}, \frac{w+\mathrm{s} \quad \mathrm{s}}{2}\right) . \] Step 2 Therefore, the midpoint of the segment between the given endpoints \( (7,5,10) \) and \( (3,1,6) \) is \[ \left(\frac{7+3}{2}, \frac{5+1}{2}, \frac{10+6}{2}\right)=(5,3,8) . \] Step 3 The radius of the sphere is the distance from this midpoint to either endpoint. Therefore, \[ r=\sqrt{(3-5)^{2}+(1-3)^{2}+(6-8 \quad 8)^{2}} \] \[ =\sqrt{12} . \] Step 4 Recall that the equation of a sphere with radius \( r \) and center \( (a, b, c) \) is given by \[ (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2} . \] Substituting the radius and center obtained above into the equation of a sphere, we therefore conclude that the equation of the given sphere is as follows. \[ (x-2)^{2}+(y-2)^{2}+(z-2)^{2}=\sqrt{12} \]