Rectangles beneath a curve? A rectangle is constructed with one side on the positive ? x-axis. one side on the pos? itive ?y-axis, and the vertex opposite the origin on the curve y = cos x, for 0 < x < ?/2. Approximate the dimensions of the rectangle that maximize the area of the rectangle. What is the area?
Solution Step 1 In this problem a rectangle is constructed with one side on the positive x-axis and one side on the positive y-axis and the vertex opposite to the origin on the curve y = cos x, for 0 < x < /2. We have to find the dimensions of the rectangle that maximize the area of the rectangle. Step 2 Let the length of the rectangle be x and the breadth of the rectangle be y . Then the area of the rectangle is A = xy Given that the requirement of the rectangle lies on the curve of y = cos x, Therefore area A = x cos x To find the dimensions which increases the area, we have to find the x-derivative of A and equate it to 0. We have A = x cos x A(x) = cos x x sin x Now equating it to 0 we get., cos x x sin x = 0 cos x = x sin x x = cos x sin x x = cot x By the above equation it is impossible to solve for x. so let us use the graph of xand cot xdrawn below
