Trig question

[lovely_nancy]

Really need help on this question :|
dont know how to start
please guide me through!

Thanks

 

[tkhunny]

Just some hints that may lead to something.

1) Draw the centers of the circles.
2) Draw the radius from the center of each circle to the two points where the band is tangent to the circle.
3) Mark the intersections for the radii and the band as right angles.

See if that gets you started.

 

[lovely_nancy]

tried but still sort of confused

 

[galactus]

Locate the centers of the circles and connect the dots to form a triangle. This triangle is equilateral with known side lengths.

Draw lines straight out from the centers to where the band touches the circles. These straight parts are easy...they are 7 mm each.

You could also extend the straight parts with a pencil and form another triangle. You can find how much of the circle is touched by the band.

The radii of the circles are given. Fortunately they are all the same. Start drawing triangles and labeling lengths and angles.

It will fall into place.

 

[lovely_nancy]

the answer im getting for the first part of the question is 2.11
is it correct?

 

[galactus]

That's not too far off. I got something slightly smaller. But, hey, maybe I erred.

For the shaded area:

Make an equilateral triangle whose vertices are the centers of the circles.

Each side of this triangle has length s=7. Where the area of an equilateral triangle is \(\displaystyle \frac{\sqrt{3}}{4}s^{2}\)

Now, find the area of each circular sector made from the angles of the triangle. Each angle is \(\displaystyle \frac{\pi}{3}\).

The formula for a circular sector is \(\displaystyle \frac{1}{2}r^{2}\theta\).

Subtract the area of the 3 sectors from the area of the equilateral triangle. This gives the shaded area.