Calculus Help

[EternalWrath]

Here I am with yet another mathematical dilemma; it just goes to show how bright the people who come here really are. ;)

OK, so you have an isosceles triangle of unknown dimensions with a circle inscribed inside. What we need to find is the dimensions of the triangle that will yield the smallest area under these conditions. After thinking about it for a bit, I've come to the conclusion that an equilateral triangle (which is also isosceles by definition) would be the solution, but I don't know how to solve it without this assumption.

Please show your method of solving this problem as well as the solution you get, and thanks for your help. :-P

[Reach]

The dimensions? As small as physically possible would yield the smallest inscribed circle? >_>

But I suppose you can consider if it's a right triangle, an isosceles (exactly two equal sides) or an equilateral.

You can solve for the radius of the inscribed circle to find:

Equilateral: R= length of a side x (root3)/6
Right: R= ab/a+b+c
Isosceles: Don't feel like writing this out

Alright, so you can set all of their areas to 1 and find the length of their sides, keeping them isosceles. Then plug them in and I get:

R for right triangle: 0.4142
R for equilateral: 0.4387
R for isosceles: 0.3051 (given the side lengths I chose)


I did this pretty quickly but whatever, it looks like the triangle should be truly isosceles, as in has exactly two sides of equal length. Which makes sense, since I didn't consider the angle here but if you continue to make it smaller you'll get a smaller inscribed circle.

[devonin]

Aaaaand, off to chit-chat with you. Help with homework, even as relatively "advanced" homework as calculus (Putting you, I assume somewhere in grade 11 or 12?) has no place in critical thinking.

Most of all when you aren't actually asking for help, but instead just a solution.

Edit: Oh, you got your answer, forget moving.