Thread: calc 3, maximize the area

1. calc 3, maximize the area

a triangular field is to be enclosed by p feet of fencing so as to maximize the area of the field. find the lengths of sides of this triangle. [Hint: Heron's formula for the area of a triangle with side lengths x, y, and z is A= square root of (s(s-x)(s-y)(s-z)), where s= 1/2 (x+y+z) is the semiperimeter.]

so, there's a function with two constraints.
i took the gradient of each.
i set gradient p = lambda gradientA + mu gradientS
this is extremely complicated though and i don't know how to solve it.

the back of the book says x=y=z=p/3

any suggestions? thank you so much

2. Originally Posted by holly123
a triangular field is to be enclosed by p feet of fencing so as to maximize the area of the field. find the lengths of sides of this triangle. [Hint: Heron's formula for the area of a triangle with side lengths x, y, and z is A= square root of (s(s-x)(s-y)(s-z)), where s= 1/2 (x+y+z) is the semiperimeter.]

so, there's a function with two constraints.
i took the gradient of each.
i set gradient p = lambda gradientA + mu gradientS
this is extremely complicated though and i don't know how to solve it.

the back of the book says x=y=z=p/3

any suggestions? thank you so much
Here's a hint: maximizing $\sqrt{s(s-x)(s-y)(s-z)}$ is the same as maximizing $s(s-x)(s-y)(s-z)$. That should make the calculations simpler.