Probem43+Sam+B,+Elie

43: 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 volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle theta is amzingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by

S=6sh-(3/2)s^2cot(θ) +[3s^2(3/2)^.5]csc(θ)

where s, the length of the sides of the hexagon, and h, the height are constants. (a)Calculate dS/dθ (b)What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h)

Note: Actual measurements of the angle in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 2 degrees.

Elie:

a) dS/dθ=-3/2s^2 (-csc^2θ)+(3s^2sqrt3/2)(-cscθcotθ) dS/dθ=3/2s^2cscθ[cscθ-sqrt(3)cotθ]

b) Set derivative equal to zero. 3/2s^2cscθ[cscθ-sqrt(3)cotθ]=0 Separate left side:

3/2s^2cscθ=0 and cscθ-sqrt(3)cotθ=0

cscθ can never equal zero, so only the second equation possible.

cscθ-sqrt(3)cotθ=0 1/sinθ=sqrt3(cos/sin) cosθ=1/sqrt3 θ= 54.74 degrees

c)S=6sh-(3/2)s^2cot(54.74) +[3s^2(3/2)^.5]csc(54.74) S=6sh-1.06s^2+3.18s^2 S=s[6h-1.06s+3.18s] S=s[6h+2.12s]

Sam- NICE ONE ELIE!!!