Application of differentiation: dimensions of prism

[Guest]

3) The cross-section of a solid prism is an equilateral triangle. The sum of the lengths of the edges of the prism is 18 cm.

a) Find the exact side length of the triangle in cm so that the volume is maximised.

b) Find the maximum volume in cm³, correct to 2 decimal places.

c) Find the exact side length of the triangle in cm so that the surface area is maximised.

d) Find the maximum surface area in cm², correct to 2 decimal places.

 

[skeeter]

assume the prism is standing on its equilateral base ...

let h = vertical edge length
s = base edge length

3h + 6s = 18
h + 2s = 6
h = 6 - 2s

V = (base area)(height)
V = [sqrt(3)/4]s²h
V = [sqrt(3)/4]s²(6 - 2s)
find dV/ds, set dV/ds = 0 and determine the value of s such that the prism has max volume, then determine that volume.

surface area ...
A = 2(base area) + 3(lateral face area)
A = [sqrt(3)/2]s² + 3sh
A = [sqrt(3)/2]s² + 3s(6 - 2s)
same drill here ... determine dA/ds and minimize.