A wire 100 inches long is to be cut into two pieces. One of the pieces will be bent into the shape of a circle and the other into the shape of a equilateral triangle. How should the wire be cut so as to maximize the sum of the area of the areas of the circle and triangle will be maximized?

Answer 1

L=total length,
x=length of piece for triangle

So
L-x=circumference of circle
At(x)=area of triangle
=√3 *x²/4
Ac(x)=area of circle
=π((L-x)/(2π))²
=(L-x)²/(4π)

Total Area, A(x)
= At(x)+Ac(x)
To find the maximum/minimum,
Equate
A'(x)=0 and solve for x.
Find A(x) = max. area.