The number in each cell is the sum of numbers in cells in two neighbor cells. Two numbers are already written in the cells. Which number needs to be written in the cell marked with x ?

The fraction can be written as the following sum:

where the numbers x, y and z are solutions of 

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by

where s, the length of the sides of the hexagon, and h, the height, are constants.

(a) Calculate .

(b) What angle should the bees prefer?

(c) Determine the minimum surface area of the cell (in terms of s and h).

Step 1. Draw an area model for a fraction with units of thirds, fourths, or fifths.

Step 2. Shade in more than one fractional unit.

Step 3. Partition the area model again to find an equal fraction.

Step 4. Write the equivalent fractions as a number sentence. (If you have written a number sentence like this one already in this homework, start over.)