Question

Prove that the greatest area of a rectangle inscribed in a triangle is one half of the area of the triangle. Use calculus

Answer 1

a triangle can be defined as a line: y = mx + b

Answer 2

the area of a rectangle from this setup is 2xy

Answer 3

ok I get the triangle being defined as a line part but wouldn't the area of a rectangle simply be xy?

Answer 4

using lagrange multipliers ....

f(x,y) = 2xy ; g(x,y) = mx - y + b

fx = 2y ; Lgx = mL ; y = mL/2
fy = x ; Lgx = -L ; x = -L

plugging these into the function g: y-mx = b

mL/2 - m(-L) = b

solve for L

3mL/2 = b

L = 2b/3m; x= -2b/3m, y = -1b/3

Answer 5

since the line only represents half the triangle .... height as y and width as 2x

Answer 6

if we want to make it general, we would need to be able to define all triangles .... at the moment this is just for right/isosolese :/

Answer 7

well I think the problem is I haven't learned lagrange variables

Answer 8

sorry lagrange multipliers

Answer 9

lagrange is a method ... im sure there are other methods just as suitable, i just cant think what theyd be at the moment :)

Answer 10

okay... im sorry but I really didn't get your process