The sum of the perimeter of a equilateral triangle and square is 10. Find the dimensions of the triangle of and square that produce a minimum total area.
Perimeter of square is 4x.
Perimeter of Equilateral is 3y.
Area of square is x^2.
Area of Equilateral triangle is [y^2*sqrt(3)]/4.
Total area is x^2 + [y^2*sqrt(3)/4].
Total perimeter is 3y + 3x, where 3y + 4x = 10.
I then solved 10 = 3y + 4x for y and got y = (10-4x)/3.
Afterward, I plugged y into the total perimeter.
Let T = total perimeter
T = x^2 + (1/4)[(10-4x)/3]^2*sqrt(3).
I then found T prime.
T' = 2x + (2x-5)/3*sqrt(3)
Let T' = 0 to find x.
I found out that x = 5/[6*sqrt(3) + 2]
I stopped here because it does not make sense to me.
Can someone get me started in the right direction?