Question

if 60 m of fencing is available for a rectangular garden, one side of which is against a barn, what areb the dimensions of the garden that will give the maximum area?

Answer 1

If the sides are x and y, with x parallel to the barn,

x+2y = 60, so
x = 60-2y

the area is

a = xy = (60-2y)y = 60y-2y^2
da/dy = 60 - 4y
max a when y=15,x=30

Answer 2

Just noticed this was for trig, not calculus. Algebraically, note that a(y) is a parabola, whose vertex occurs where y=60/4 = 15.