Awire 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?

1. Divide wire b in parts x and b-x. 

2. Bend the b-x piece to form a triangle with side (b-x)/3

There are many ways to find the area of the equilateral triangle. One is by the formula A= 
A=

Another way is apply the formula A=1/2*base*altitude,
where the altitude can be found by applying the pythagorean theorem on the triangle with hypothenuse (b-x)/3 and side (b-x)/6

3. Let x be the circumference of the circle.

 

so 

Area of circle = 

4. Let f(x)=

be the function of the sum of the areas of the triangle and circle.

5. f(x) is a minimum means f'(x)=0

f'(x)==0







6. So one part is  and the other part is b-

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