The sum of the perimeters of an equilateral triangle and a square is 10. find the dimensions of the triangle and the square that produce a minimum total area.

X - side of an equilateral triangle,  y - side of a square;
3 x + 4 y = 10
4 y = 10 - 3 x
b = ( 10 - 3 x ) /4
Total area:
A = x ²√3/4 +y² = x²√3/4 + ( 10- 3 x )²/16 = x²√3/4 + 9 x²/16 - 15 x/4 + 25/4
A` = x√3/2 + 9 x/8 - 15/4
A` = 0
4 x√3 + 9 x = 30
15.92 x = 30
x = 30 : 14.92 ≈ 1.88
y = (10 - 1.88 ) : 4 ≈ 2.03
dimensions are x = 1.88 ( triangle ) and y = 2.03 ( square )