Question

a wire 60cm long is to be cut into two pieces, one of which is to be bent into the shape of a square and the other of which is to be bent into an equilateral triangle. find how the wire should be cut so that the combined area of the square and equilateral is minimum

Answer 1

If the square has side s and the triangle has side t, then
4s+3t = 60
The area is
a = s^2 + √3/2 t^2
Now, since t = 20 - 4/3 s,
a = s^2 + √3/2 (20 - 4/3 s)^2
da/ds = 2s - 16(15-s) / 3√3
so a is minimum when da/ds = 0

Answer 2

g