What is the blue area in the graphic at the following link:
drive<dot>google<dot>com/file/d/1w4WF6LfpJB34pkmeRnN5EJsOPuk-ZQUS/view
let O = (0,0)
Then the circle with center at O is x^2+y^2 = 1
The circle with diameter OB is (x - 1/2)^2 + y^2 = 1/4
Now let F be (x,y). Then
C is at (x,√(1-x^2))
F is at (x,√(1/4 - (x - 1/2)^2)
and since CF = 2x, we have
√(1-x^2) - √(1/4 - (x - 1/2)^2 = 2x
Now just solve for x^2, the area of the blue region
I am able to see your diagram
Here would be my approach.
Place the figure on the x-y grid with O at the centre.
Then B is (1,0) and A is (-1,0) , with the equation of the large circle is
x^2 + y^2 = 1
the equation with OB as diameter is
(x - 1/2)^2 + y^2 = 1/4
and the equation with AO as diameter is
(x + 1/2)^2 + y^2 = 1/4
Let the coordinates of D be (a,b) and the width of the square as c
then for the other points on the square we have:
C(a+c, b) , E(a, b-c) , and F(a+c, b-c)
Here comes the fun part:
Both D and C satisfy the equation x^2 + y^2 = 1
a^2 + b^2 = 1 and (a+c)^2 + b^2 = 1
subtract them:
(a+c)^2 - a^2 = 0
a^2 + 2ac + c^2 -a^2 = 0
2ac = -c^2
c = -2a <---- easy so far
E lies on (x+1/2)^2 + y^2 = 1/4
(a+1/2)^2 + (b-c)^2 = 1/4
(a+1/2)^2 + (b + 2a)^2 = 1/4
a^2 + a + 1/4 + b^2 + 4ab + 4a^2 = 1/4 , but a^2 + b^2 = 1
1 + a + 4ab + 4a^2 = 0
So we have
a^2 + b^2 = 1 and
1 + a + 4ab + 4a^2 = 0
Not looking forward to solving this, so let's go to Wolfram:
https://www.wolframalpha.com/input/?i=solve+1+%2B+a+%2B+4ab+%2B+4a%5E2+%3D+0+%2C+a%5E2+%2B+b%5E2+%3D+1
We get multiple answers, e.g. there would be a reflection of the whole
thing in the x-axis, etc
looking at the answers and the diagram, a plausible solution is
a = -0.262478 , b = 0.964938
which would make c = -2a = .524956
and the area is c^2 = appr .2755...
better check those calculations, chances of error are great.
Oops. I haven't checked my solution against Reiny's, but I must note that the area of my square is 4x^2, not x^2.
But I'm sure you caught that.
Thanks for your help, guys. This problem's been nagging me for a while. The area of the square was 4x², once you solved for x............
I apologize, but as an AI text-based bot, I am unable to access external links or view specific graphics. However, I can assist you by explaining how to find the answer to your question visually.
To determine what the blue area represents in the graphic you mentioned, you can follow these steps:
1. Open the provided link in your web browser to view the graphic.
2. Look for any labels, legends, or descriptions accompanying the image. This could provide information about what the blue area represents.
3. Examine the graphic itself. Pay attention to any patterns, colors, or symbols used in the image.
4. If there are no explicit labels or legends, try to infer the meaning of the blue area based on its context within the graphic. Consider any related elements or information presented alongside it.
5. If the graphic includes a title or caption, read it carefully as it may provide insights into what the blue area represents.
6. If the graphic is part of an article, blog post, or webpage, read the surrounding text to gain further context or explanation.
By following these steps and examining the provided graphic and its surroundings, you should be able to determine what the blue area represents.