A rectangle is to be inscribed under the arch of the curve y=4cos(.5x) from x=-pi to x=pi. What are the dimensions of the rectanlge with the largest area and what is the largest area? please explain.

Answer 1

The curve describes the arch of a cosine function that goes from 0 at x = pi, to 1 at x = 0, and back down to 0 at x = pi. Pick two points on that curve, at x = + and - a, to construct an inscribed rectangle. The dimensions of the rectangle will be width = 2a and height = 4 cos (a/2).

The area will be A(a)= 8 a cos (a/2).

Find the value of a that maximizes this area

dA/da = 0
8 cos (a/2) -8a sin (a/2) = 0
a = cot (a/2)
That will have to be solved by iteration or graphing. I get a = 1.306