Official Solution: A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only

B. II only

C. III only

D. I and II only

E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally,

the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,

since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is \(0 \lt area \le 2\). So, I and II are possible values of the area. (Else you can notice that the area of the circle is \(\pi r^2=\pi \approx 3.14\) and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Answer: D

i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?

let a = side of square...

a = sqrt(2).

Area = pi*a^2 = 2*pi.