3-D Inequalities

[suicoted]

I'm having a tough time working with 3-D, 3 axes. I'm not sure how to approach these:

Q: Write inequalities to describe the sets.

1. The solid cube in the first octant bounded by the coordinate planes and the planes x=2, y=2, and z=2.
2. The half-space consisting of the points on and below the xy-plane.
3. The upper hemisphere on the sphere of radius 1 centered at the origin.
4. The (a) interior and (b) exterior of the sphere of radius 1 and center (1,1,1).

Thanks.

 

[stapel]

1) It's solid, which actually makes things easier sometimes: you're not having to come up with the equation of a "surface". Think about (or draw) the situation. If the solid is in the first octant, can any of the variables take on negative values? So what inequalities describe the values for x, y, and z? (The answer will be a set of three inequalities.)

2) In the x,y-plane, are there any limits on the values of x or y? What is the value of z when you're "on" the x,y-plane? Below the x,y-plane, what are the values for z? (The answer will be an inequality for one of the variables.)

3) What is the equation of the sphere centered on the origin with r = 1? Solve this for "z=", and then take the appropriate "half". (Hint: Solving will produce a "±". You need half of this.) (Note: I'm not sure if this is supposed to be a solid or a surface.)

4) What is the equation for the sphere? What inequality would mean "outside"? What would mean "inside"?

Eliz.

 

[suicoted]

1) It's solid, which actually makes things easier sometimes: you're not having to come up with the equation of a "surface". Think about (or draw) the situation. If the solid is in the first octant, can any of the variables take on negative values? So what inequalities describe the values for x, y, and z? (The answer will be a set of three inequalities.)

2) In the x,y-plane, are there any limits on the values of x or y? What is the value of z when you're "on" the x,y-plane? Below the x,y-plane, what are the values for z? (The answer will be an inequality for one of the variables.)

3) What is the equation of the sphere centered on the origin with r = 1? Solve this for "z=", and then take the appropriate "half". (Hint: Solving will produce a "±". You need half of this.) (Note: I'm not sure if this is supposed to be a solid or a surface.)

4) What is the equation for the sphere? What inequality would mean "outside"? What would mean "inside"?

Eliz.

Thanks, but can anyone clarify 'half-space' for #2...

For #1, I got 0<=x<=2,0<=y<=2,0<=z<=2, because the first octant is all positive numbers.

For #3, if it is just the upper hemisphere, am I just concerned with the z?

 

[pka]

Any plane determines two half-spaces.
Each half-space is one side of the plane.
What are the coordinates of all points on or below the xy-plane?

The set \(\displaystyle \{ (x,y,z):x^2 + y^2 + z^2 < r^2 \}\) is the interior of a sphere centered at the origin with radius r.