let the base be 2x ,(if x = 1/2 the base)
and let each of the equal sides be y
so we know 2x + 2y = 30
x+y = 15, or y = 15-x
let the height of the triangle be h
then h^2 + x2 = y^2
h^2 + x^2 = (15-x)^2
h^2 + x^2 = 225 - 30x + x^2
h = √(225-30x)
area = (1/2)base x height
= (1/2)(2x)(√(225-30x) )
= x (225-30x)^(1/2)
d(area) = x(1/2)(225-30x)^(-1/2) (-30) + (225-30x)^(1/2)
=0 for a max of area
(225-30x)^(1/2) = 15x/(225-30x)^(1/2)
15x = 225-30x
45x = 225
x = 5
then h = √(225-150) = √75
and since x+y = 15
y = 10
so each side of the triangle must be 10 yds
and the largest area is (1/2)(10)(√75)
Makes sense that the triangle of largest area would be an equilateral triangle, just like the largest quad would be a square, etc.
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