Step 1 :Given the function \(h(r)=3 r^{2}-3 r+1\), where \(r\) is the number of rings, and \(h\) is the number of hexagons.
Step 2 :For part A, we substitute \(r=10\) into the function to find the expected number of hexagons when the honeycomb has 10 rings.
Step 3 :\(h(10)=3(10)^{2}-3(10)+1 = 271\)
Step 4 :So, the expected number of hexagons when the honeycomb has 10 rings is \(\boxed{271}\) hexagons.
Step 5 :For part B, we set \(h(r)=29107\) and solve for \(r\). This gives us a quadratic equation \(3r^{2}-3r-29106=0\).
Step 6 :We calculate the discriminant \(D = b^{2}-4ac = (-3)^{2}-4*3*(-29106) = 349281\).
Step 7 :Then we find the roots of the equation using the quadratic formula \(r = \frac{-b \pm \sqrt{D}}{2a}\).
Step 8 :This gives us two roots, \(r1 = \frac{3 + \sqrt{349281}}{6} = 99\) and \(r2 = \frac{3 - \sqrt{349281}}{6} = -98\).
Step 9 :Since the number of rings cannot be negative, we discard \(r2\).
Step 10 :So, the number of rings required to create exactly 29,107 hexagons is \(\boxed{99}\) rings.