Calc III, what point could this be?

In summary, if x=0 and y=2, then the solution set is the line segment x=0, y=2,-\sqr{12}\leq{z}\leq\sqrt{12}. If you are interested in the "top most" point, then (0,2,\sqrt{12}) should be it.
  • #1
mr_coffee
1,629
1
Hello everyone, I'm trying to figure out this point on the sphere. The orginal equation is this: x^2 + y^2 + z^2 ≤ 16; 2 ≤ y ≤ 4;

Here is the picture:
http://img133.imageshack.us/img133/5527/point1eo.jpg

If you look where it shows a picture of the sphere and i have an arrow pointing to a point on top of the circle. Has coordinates (0,2, ? ). I"m trying to figure out the point where that ? is. I know the x and y coordinates havn't changed at all, only the z coordinates. Sorry its alittle messy. :bugeye: I know the point below it is (0,2,0). Thanks.
 
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  • #2
First off, you're dealing with inequalities, not equations.
Secondly, the solution set to the two given inequalities is a REGION, a segment of the sphere, not just a single point on the sphere.
 
  • #3
I realize the soution set is a region not a single point on my sphere, my question was, what the point was, not what the whole solution set was.
 
  • #4
It is rather unclear what you're after, but:
Well, if x=0 and y=2, then the solution set is the line segment
[tex]x=0, y=2,-\sqr{12}\leq{z}\leq\sqrt{12}[/tex]
If you are interested in the "top most" point, then [tex](0,2,\sqrt{12})[/tex] should be it.
 
  • #5
[tex]x^2+y^2+z^2\le 16[/tex]
describes the points inside and on a sphere of radius 4, centered at the origin.

[tex]1\le y\le 4[/tex]
describes points between and on the two planes y= 2 and y= 4

Of course, the plane y= 4 is tangent to the sphere at (0, 4, 0).

The plane y= 2 crosses the sphere in the circle
[tex]x^2+ 2^2+ z^2= 16[/tex]
or [tex]x^2+ z^2= 12[/tex].

That is a circle in the plane y= 2, centered at (0,0,0) with radius [tex]2\sqrt(3)[/tex].
In particular, the point you are apparently looking for is (0, 2, 2√(3)).

The set of all points described by the inequalities are the points inside or on the sphere "beyond" (whatever direction the y-axis is in!) y= 2.
 
  • #6
Thanks, how did you figure out it was [tex](0,2,\sqrt{12})[/tex] for z? My professor requested we figure out that point on the circle that's why its such an odd question. He said, to figure it out all you would use is basic aglebra.
 
  • #7
Thanks Halls, I think your answer is right! It makes sense anyways!
 
  • #8
Okay, I didn't read your post closely enough; you did mention a circle somewhere...
Anyhow, my answer is the same as HallsofIvy's
 
  • #9
yep thank you both for the help
 

1. What is the purpose of studying Calc III?

The purpose of studying Calc III is to understand and apply advanced concepts in multivariable calculus, such as vector calculus, partial derivatives, multiple integrals, and vector fields. These concepts are essential for further studies in fields such as physics, engineering, and economics.

2. What topics are covered in Calc III?

Calc III covers topics such as vectors, vector-valued functions, partial derivatives, multiple integrals, line integrals, surface integrals, and vector calculus. Students will also learn about the applications of these concepts in real-life problems.

3. How is Calc III different from Calc I and Calc II?

Calc III builds upon the concepts learned in Calc I and Calc II, but focuses on functions with multiple variables rather than just one. This includes learning about vectors, vector-valued functions, and vector calculus. It also involves more complex integration techniques, such as multiple integrals and line integrals.

4. What are some common applications of Calc III?

Some common applications of Calc III include physics, engineering, economics, and computer graphics. For example, understanding vector calculus is essential in analyzing the motion of objects in three-dimensional space, and multiple integrals are used to calculate the volume and surface area of complex shapes.

5. How can I succeed in Calc III?

To succeed in Calc III, it is important to have a strong understanding of the concepts taught in Calc I and Calc II. Practice and review regularly, and make use of resources such as textbooks, online tutorials, and practice problems. It is also helpful to attend lectures and participate in class discussions. Seeking help from the professor or a tutor when needed can also greatly improve understanding and success in the course.

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