What is the area between two circles with a diameter of 10m?

[skybox]

Homework Statement

Homework Equations

The area of a circle:
$$A_c = \pi r^{2}$$

The Attempt at a Solution

I know that the diameter of the oval shape is 10m since the problem says that it touches the circumference of the center of each circle. I am not sure how to approach the problem now. Any hints would be great.

Thanks

 

[spaderdabomb]

Well we have two equations to work with
$$x^2 + y^2 = 10^2$$
$$(x-10)^2 + y^2 = 10^2$$

Theses give the equations of the two circles if we place the origin at the center of the circle on the left. Next I would advise finding the intersection points. Hopefully you can continue to work on it some more and determine the area using these equations.

 

[spaderdabomb]

If you get stuck again, I imagine it will be finding the area once you get those two points. I would highly recommend reading this page on wikipedia http://en.wikipedia.org/wiki/Spherical_cap. You can obtain the equation for spherical caps. I'm not sure if this is the way your instructor wants to do it or not though. It is possible to derive it yourself using the equations, but this will make it much less painless =)

 

[verty]

This is very much like an olympiad question, so much so that I think we should refrain from explaining how to solve it and leave it as a challenge for each reader to solve.

As a hint, there is an easier way to find the points of intersection that uses only geometry, and to solve it, only geometry and some very elementary trig knowledge is needed.

 

[spaderdabomb]

This is very much like an olympiad question, so much so that I think we should refrain from explaining how to solve it and leave it as a challenge for each reader to solve.

As a hint, there is an easier way to find the points of intersection that uses only geometry, and to solve it, only geometry and some very elementary trig knowledge is needed.

I find this to be easier than whatever geometry you're using

$$x^2 - 20x + 100 + y^2 = x^2 + y^2$$
$$x = 100/20 = 5, y = \pm 5\sqrt{3}$$

haha, but its always nice to have multiple ways to solve things. Of course, the next part of the problem is really the hard part =p

 

[verty]

In case it is needed, here is a second hint: Divide the area down the middle, find half, then double it.

This way I know Skybox has ample hints to solve it. So I hope we can now leave it unanswered, thank you.

 

[skybox]

Thanks all for the replies. I think I have an idea on how to solve the problem with all the hints given. I will not post the solution once I do come to a solution. Thanks again!