Please answer this precalc question and explain?

Question

what is the area of the small region with the red dot in it in the diagram at the link provided. give answer correct to 3 decimal places THX sOOOOO MUCH!!! http://i128.photobucket.com/albums/p199/av611/hoil.jpg

Answer 1

I can tell you what you need to do, but I don't remember all the formulas for everything - you can ask someone else later if you don't want to look it up yourself.

From the centers of each of the circles, draw a radius toward the center of each of the other circles. A radius of 6 will meet up with a radius of 4 to form a line of length 10. A radius of 6 will meet up with a radius of 2 to form a line of length 8. A radius of 4 will meet up with a radius of 2 to form a line of length 6. The lines connect to form a 6-8-10 right triangle (your basic 3-4-5, with sides all multiplied by 2).

You need to use trig to figure ALL the angles. The 90 degree angle will be opposite the side of length 10, inside the circle of radius 2. (I don't remember the other angles inside a 3/4/5 triangle).

Find the area of the triangle.
Find the area of the part of each circle determined by the angle between the radii you have drawn. (I don't remember that formula either.)

Subtract the area of the 3 circle sections from the area of the triangle and you have your answer.

That should give you a start, anyhow. Good luck.

Answer 2

use the midpoint of each circle, and connect them... it was form a 3:4:5 right triangle. with sides length 6:8:10. Of which the area of that is easy to find.

Then need to subtract the area inside each cicle within the triangle.

one cicle will have the 90 degree angle, smallest one, so that area is 1/4th the whole circle.

the other 2 cicles, use the sides of the triangles, 6:8:10, to find the measure of the angles.

so for the large circle radius 6.. the angle would be sinx=6/10 or cosx=8/10, about 38.....

and for the medium circle, radius 4, the angle would be cosx=6/10 or sinx=8/10. about 53....

then using tha angles, you get a percentage of the total circles area by /360

so the area you are looking for is =
Area of triangle =24
- 90/360 X 2 X 2 X pi
-53/360 X 4 X 4 X pi
- 38/360 X 6 X 6 X pi

not sure the exact number, but you can work the rest out...