(a) Using Lagrange multipliers, find the area of the largest triangle inscribed in a circle
of radius R.
(b) Using Lagrange multipliers, find the volume of the largest tetrahedron (= pyramid
with triangular base) inscribed in the unit sphere x^2+y^2+z^2 =1.
For part a I was able to figure out that the largest triangle is an equilateral triangle with side length sqrt(3)R and a height of (3/2)R, giving an overall area of (1/2)(sqrt(3)R)((3/2)R) = (3sqrt(3)R^2)/4
I am really stuck on part 2 though! I know I can use the information from part A given that the volume of a tetrahedron is (1/3) times the area of the base times the height, but I am unsure of where to start the problem. Any help would be appreciated!
of radius R.
(b) Using Lagrange multipliers, find the volume of the largest tetrahedron (= pyramid
with triangular base) inscribed in the unit sphere x^2+y^2+z^2 =1.
For part a I was able to figure out that the largest triangle is an equilateral triangle with side length sqrt(3)R and a height of (3/2)R, giving an overall area of (1/2)(sqrt(3)R)((3/2)R) = (3sqrt(3)R^2)/4
I am really stuck on part 2 though! I know I can use the information from part A given that the volume of a tetrahedron is (1/3) times the area of the base times the height, but I am unsure of where to start the problem. Any help would be appreciated!