Help with critical points and extrema problem (calculus)

Question

The surface area of a cell in a honeycomb is given by the following, where h and s are positive constants and 𝜃 is the angle at which the upper faces meet the altitude of the cell (see figure).

S = 6hs + 3s²/2 * (sqrt3 - cosθ)/sinθ

Find the angle 𝜃 in radians (𝜋/6 ≤ 𝜃 ≤ 𝜋/2) that minimizes the surface area S.

Answer 1

S(θ) = 6hs + 3s² * (((√3) - cosθ) / sinθ)

S'(θ) = (1 - (√3)cosθ) / sin²θ [taking the derivative with respect to θ]

Set S'(θ) = 0 and solve for θ to find critical point(s)

1. 1 - (√3)cosθ = 0 [set the numerator equal to 0]
2. cos^-1((√3)/3) = θ
3. θ = 0.955 radians

θ is within (𝜋/6 ≤ 𝜃 ≤ 𝜋/2), so that's our answer.

Hope this was helpful.