Question 13
9 pts
If you look at the cross-section of a honeycomb, you see a pattern of tessellating hexagons. This pattern starts with one hexagon surrounded by six hexagons. Those are surrounded by a third ring of twelve hexagons, and so on. The total number of hexagons in a honeycomb can be modeled by the function $h(r)=3 r^{2}-3 r+1$, where $r$ is the number of rings, and $h$ is the number of hexagons.
Rings of a Honeycomb
A. Find the expected number of hexagons when the honeycomb has 10 rings. hexagons
B. Calculate the number of rings required to create exactly 29,107 hexagons. rings

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.