Use Lagrange multipliers to prove that the triangle with...

Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. Hint: Use Heron's formula for the area: A = sqrt{s(s - x)(s - y)(s - z)} where s = p/2 and x, y, z are the lengths of the sides.

Transcript

00:01 All right, we're going to find the triangle that has the maximum area that has a fixed perimeter peak.

00:09 Okay, so usually if we want to talk about the area of the triangle, we're going to use areas one -half base times height.

00:16 But we don't know the base or the height.

00:19 All we know is the perimeter is p, so we can't use that.

00:23 And then there's this formula, area equals 1 -5 -a -b -sign c, where this is side -angle side.

00:31 The c is between the a and b.

00:34 Okay, we can't do that.

00:35 We don't know any angles in there.

00:36 So then they say, okay, we've got to use this heron's formula, which goes semi -parameter, semi -parameter minus one side, semi -parameter minus the other, semi -parameter minus the third, and then multiply them all together and take the square root.

00:53 So that's the area formula we're going to use.

00:56 Okay, now the trick here is we know that the semi -parimeter that means the half of the parameter around s over s is p over 2.

01:15 Okay, so that's a semi -parimeter.

01:17 So s is not the variable here.

01:20 It's the a and b and c.

01:21 They're x and y and z.

01:23 So i'm going to switch to that.

01:25 So the area is the square root of s.

01:31 Wait, i know what s is.

01:33 I'm going to put that in.

01:36 P over two.

01:37 P over 2 minus x p over 2 minus y p over 2 minus z okay so we're going to maximize that okay subject to x plus y plus z equals p and once you get it to there it's not nearly as bad as it looked like it was going to be to start with all right so first of all i'm going to use the fact that if i maximize the function, then i'm maximizing the square root...

Question

Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. Hint: Use Heron's formula for the area: A = sqrt{s(s - x)(s - y)(s - z)} where s = p/2 and x, y, z are the lengths of the sides.

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