A wire b units long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part?

if you notice the picture below

if we take "x" amount for the triangle, then the circle will be using "b-x", since the wire is "b" long

now, that's just their perimeter only

anyhow. ..the sum of their lengths will then be [tex]\bf A_t=\cfrac{x^2\sqrt{3}}{36}+\cfrac{(b-x)^2}{4\pi }[/tex]

so... take the derivative of that areas sum, and check the critical points for a minimum, using the first-derivative test

keeping mind that, "b" is a constant, since that matters for taking the derivative

A wire b units long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a​ minimum, what is the length of each​ part?

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A wire b units long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part?