Prove that the portion of a sphere of radius R seen by an observer located at a distance d above

Prove that the portion of a sphere of radius R seen by an...

Key Concepts

Spherical Cap

A spherical cap is the portion of a sphere cut off by a plane. Its study involves understanding the geometry of the sphere and how such segments are formed. Problems involving spherical caps often require calculating their surface area and relating the cap’s dimensions to the sphere’s radius and other geometric parameters.

Surface Area Formula for a Spherical Cap

The surface area of a spherical cap is given by the formula A = 2?R h, where R is the radius of the sphere and h is the height of the cap. This formula is derived by integrating over the curved surface of the sphere and is used to quantify the size of the portion of the sphere visible when cut by a plane.

Application of the Pythagorean Theorem in Spherical Geometry

The Pythagorean theorem is used in spherical geometry to relate the distances within right triangles that can be formed in or around the sphere. By applying the theorem to these triangles, one can derive relationships between the sphere’s radius, the observer's distance from the sphere, and the height of the spherical cap, thus allowing a deeper analysis of visibility and projection problems.

Transcript

00:01 All right, so in this question, we're going to be looking at basically arc length, surface area, that type of thing.

00:11 All right, so in this question, what we're given is that the area of, so a cap, like the north pole, right, that the cap of the north pole, that top of the sphere has surface area of 2 pi r .h.

00:28 And we have to prove that the portion of the sphere of a radius are that somebody standing this d distance away from above the north pole has the area of 2 pi d r squared over d plus r right so we're trying to prove that area equals 2 pi d r squared over d plus r, right? okay.

01:07 So, and keep in mind, dr here is not the derivative.

01:12 It's literally the distance here, right? so this is the figure that we're given, and so from this we have to apply the pythagorean theorem, theorem, right? and this is really the important part of the, the important parts of the triangle.

01:30 So what you can start out with is, know that hypotenuse, so let's see, the pythagorean theorem tells us that a squared plus b squared equals c squared, right? where a, b, and c are the sides.

01:45 A, b are the sides, and c is the, is basically the hypotenuse.

01:54 So what do we know from there? well, from there, on this side we have d plus h, right? and on this side, we have r minus h.

02:07 And both of those together equal c right so we can plug those in and k is going to be a and r is going to be b so if you plug those in we'll get k squared plus r squared equals d plus h um plus r minus h all squared and these h is will then cancel out and then you'll be left with k squared plus r squared equals d plus r squared right um so from there, all we have to do is now get rid of k.

02:54 And, yeah, so we have to get rid of k somehow, right? so what we can do there is, it's not k minus 9, that's k is a.

03:11 And r here is b.

03:13 So we have to get rid of this k right here because that will make it easier for us to do algebra.

03:18 So if we look at this triangle right here, just this one, then what we can tell from that is we can once again use a pythagorean theorem.

03:32 Right? a squared plus b squared equals c squared.

03:35 In this case, a and b will be k squared and x squared and d plus h squared and x squared.

03:45 Right, because you have this x here, that's this line here, and d plus h is that line.

03:51 Your hypotenuse is basically k squared so you can take that and you can replace it here and then you'll get d plus h squared plus x squared plus r squared equals d uh sorry d plus r squared right so now what do we do now we can't have this x here really we want to have that x um we want to have that x basically in terms of r and h so what we can do is, finally, we can take this triangle here, this one.

04:31 So now if we take that triangle, then it becomes really easy.

04:35 Obviously, what we have to do is apply the pythagorean theorem once again.

04:40 In this case, your a squared will be x squared.

04:44 Your b squared will be r minus h squared.

04:47 So x squared plus r minus h squared equals r squared.

04:52 Right? so from there, you can say x squared equals r squared minus r minus h squared.

05:04 And then you can substitute that back into here.

05:07 And then you will get d plus h squared plus r squared plus r squared plus minus r minus h squared all equals d plus r...