Volume of tetrahedra formed from coordinate and tangent planes

In summary, the volume of the tetrahedron, T, formed by the coordinate planes and the tangent plane to P is independent of the point (r,s,t).
  • #1
Juggler123
83
0
I have that P is the tangent plane to the surface xyz=a[tex]^{3}[/tex] at the point (r,s,t). I need to show that the volume of the tetrahedron, T, formed by the coordinate planes and the tangent plane to P is indepedent of the point (r,s,t).

I have found that P is;

[tex]\frac{x}{r}[/tex] + [tex]\frac{y}{s}[/tex] + [tex]\frac{z}{t}[/tex] = 3

A know that the volume of a tetrahedron is giving by 1/3(area of base [tex]\times[/tex] height)

But I just can't picture what this looks like, as far as I can see the volume of T has to be dependent of the point (r,s,t).

Any help anyone could give would be great! Thanks.
 
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  • #2
Juggler123 said:
I have that P is the tangent plane to the surface xyz=a[tex]^{3}[/tex] at the point (r,s,t). I need to show that the volume of the tetrahedron, T, formed by the coordinate planes and the tangent plane to P is indepedent of the point (r,s,t).

I have found that P is;

[tex]\frac{x}{r}[/tex] + [tex]\frac{y}{s}[/tex] + [tex]\frac{z}{t}[/tex] = 3

A know that the volume of a tetrahedron is giving by 1/3(area of base [tex]\times[/tex] height)

But I just can't picture what this looks like, as far as I can see the volume of T has to be dependent of the point (r,s,t).

Any help anyone could give would be great! Thanks.

You would expect the volume to depend on r,s, and t. But if you work it out you will find that it doesn't. Just write the equation of the tangent plane, find its intercepts and the corresponding volume. Here is a picture with a = 1 of one of the tangent planes to help you visualize it:
pyramid.jpg
 
  • #3
Thanks for that, think I might be starting to understand this a little bit more now.

Right I have that

[tex]\frac{x}{r}[/tex] + [tex]\frac{y}{s}[/tex] + [tex]\frac{z}{t}[/tex] = 3

and so this plane intersects the coordinate planes at x=3r, y=3s and z=3t but all of these points you know that xyz=a[tex]^{3}[/tex] so is right to then say that the intersects occur at x=3a[tex]^{3}[/tex], y=3a[tex]^{3}[/tex] and z=3a[tex]^{3}[/tex].

Hence the volume of T is given by [tex]\frac{9a^{9}}{2}[/tex]
 
  • #4
Almost right. But check the plane intercepts again. For example, 3r doesn't equal 3a3.
 
  • #5
Right think I've got it this time.

The intercepts are at x=3r, y=3s and z=3t.

Now 3r=[tex]\frac{3a^{3}}{st}[/tex], 3s=[tex]\frac{3a^{3}}{rt}[/tex] and 3z=3r=[tex]\frac{3a^{3}}{rs}[/tex]

Hence the volume of T is given by [tex]\frac{9a^{3}}{2}[/tex]
 
  • #6
Sorry that should say 3t=[tex]\frac{3a^{3}}{rs}[/tex]
 
  • #7
Juggler123 said:
Hence the volume of T is given by [tex]\frac{9a^{3}}{2}[/tex]

Looks good.
 
  • #8
I get the feeling that this can be proved with only geometric considerations with eyes closed...
 

Related to Volume of tetrahedra formed from coordinate and tangent planes

1. What is a tetrahedron?

A tetrahedron is a three-dimensional polyhedron with four triangular faces. It is a type of pyramid with a triangular base and three triangular faces that meet at a point called the apex.

2. What are coordinate planes and tangent planes?

A coordinate plane is a two-dimensional plane that is defined by two perpendicular axes, typically represented by the x and y axes. Tangent planes, on the other hand, are planes that touch a curved surface at only one point and are perpendicular to the surface's normal vector at that point.

3. How are tetrahedra formed from coordinate and tangent planes?

Tetrahedra can be formed by taking four points on a coordinate plane and connecting them with lines to form four triangles. These triangles can then be folded or rotated to create a three-dimensional shape. Tangent planes can also be used to form tetrahedra by intersecting them with the coordinate plane and creating triangular faces.

4. What is the relationship between volume and tetrahedra formed from coordinate and tangent planes?

The volume of a tetrahedron formed from coordinate and tangent planes can be calculated using the formula V = (1/3) * base area * height. The base area is the area of the triangle formed by the coordinate plane, and the height is the distance from the apex of the tetrahedron to the base. This formula can be used to find the volume of any tetrahedron, regardless of how it is formed.

5. What are some applications of tetrahedra formed from coordinate and tangent planes in science?

Tetrahedra formed from coordinate and tangent planes have several applications in science, including computer graphics, crystallography, and geological modeling. They are also used in physics to study the behavior of liquids and gases, as well as in engineering for structural analysis and design.

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