- (a) To find the smallest rectangle with sides parallel to the x- and y-axes, you need a rectangle
formed by vertical and horizontal tangents as shown in the figure. The vertical tangents are at the
x-intercepts, x = ±3. The horizontal tangents are at the points where y (not r) is a maximum. You
need, therefore, to maximize
Use the calculator to find that when θ = 0.7854. Therefore, y = 1.414, so the desired
rectangle has dimensions 6 × 2.828.
(b) Since the polar formula for the area is the area of R (enclosed by r) is
which is 14.137.
