Optimization questions to get ready for a test

In summary, the question asks for the area of a isosceles triangle with the largest possible area that can be inscribed in a circle of given radius. The answer is found by using the equation for the area of a triangle, which depends on the lengths of the sides and the angle between them.
  • #1
3
0
I was doing some optimization questions to get ready for a test. I came across one that stumped me. The question was "Find the dimensions of the isosceles triangle of the largest area that can be inscribed in a circle of radius r".

My approach was:
let y be the base of triangle
let x be the two equal sides of triange.

A(t) = (bh)/2
= (y*(x^2-(y/2)^2)^1/2))/2

I can't find the equation to relate the radius to the area. help?
 
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  • #2
Let the angle between the two longer sides be t, now imagine instead of drawing lines from the short edge's ends to the perimeter, you drew the lines to the centre of the circle, the angle these shorter sides (of length r) create would be 2t. There is a formula for finding the area in terms of sides and angles between sides, and t depends on the lengths chosen originally. This might help you.
 
  • #3
matt grime said:
Let the angle between the two longer sides be t, now imagine instead of drawing lines from the short edge's ends to the perimeter, you drew the lines to the centre of the circle, the angle these shorter sides (of length r) create would be 2t. There is a formula for finding the area in terms of sides and angles between sides, and t depends on the lengths chosen originally. This might help you.

What do you mean by "now imagine instead of drawing lines from the short edge's ends to the perimeter, you drew the lines to the centre of the circle, the angle these shorter sides (of length r) create would be 2t. "?

the formula for the area in terms of the sides of a triangle is

(s(s-a)(s-b)(s-c))^1/2

where s is (a+b+c)/2
and a,b,c are the side lengths.

is that it?
 

1. What is optimization?

Optimization is the process of finding the best solution to a problem within a given set of constraints. It involves maximizing or minimizing a certain objective function while adhering to any limitations or restrictions.

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