hw asst  24

024.

Applied Calculus II

07-28-2008

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14:52:18

Query problem 7.2.52 (was 7.2.48) identify quadric surface z^2 = x^2 + y^2/2.

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RESPONSE -->

z^2 = x^2 + y^2/2

x^2 + y^2/2 - z^2 = 0

this is a hyperboloid of two sheets with the y-axis as its axis

confidence assessment: 1

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14:57:58

What is the name of this quadric surface, and why?

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RESPONSE -->

I've changed my mind. I think that it is an elliptic cone with its axis on the z-axis because this is the variable whose coefficient is negative.

my equation x^2 + y^2/2 - z^2 = 0 fits the equation of of an elliptic cone. the traces in the coordinate planes parallel to the z-axis are intersecting lines.

confidence assessment: 2

Good. Compare with the following:

** If z = c, a constant, then x^2 + y^2/2 = c^2, or x^2 / c^2 + y^2 / (`sqrt(2) * c)^2 = 1. This gives you ellipse with major axis c and minor axis `sqrt(2) * c. Thus for any plane parallel to the x-y plane and lying at distance c from the x-y plane, the trace of the surface is an ellipse.

In the x-z plane the trace is x^2 - z^2 = 0, or x^2 = z^2, or x = +- z. Thus the trace in the x-z plane is two straight lines.

In the y-z plane the trace is y^2 - z^2/2 = 0, or y^2 = z^2/2, or y = +- z * `sqrt(2) / 2. Thus the trace in the y-z plane is two straight lines.

The x-z and y-z traces show you that the ellipses in the 'horizontal' planes change linearly with their distance from the x-y plane. This is the way cones grow, with straight lines running up and down from the apex. Thus the surface is an elliptical cone. **

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You'll be able to tell to what the following refer:

** The xz trace consists of the y = 0 points, which for z^2 = x^2 + y^2/2 is z^2 = x^2 + 0^2/2 or just z^2 = x^2.

The graph of z^2 = x^2 consists of the two lines z = x and z = -x in the yz plane. **

** If z = 2 then z^2 = x^2 + y^2/2 becomes 2^2 = x^2 + y^2 / 2, or x^2 + y^2 / 2 = 4.

This is an ellipse. If we divide both sides by 4 we can get the standard form:

x^2 / 4 + y^2 / 8 = 1, or x^2 / 2^2 + y^2 / (2 `sqrt(2))^2 = 1.

This is an ellipse with major axis 2 `sqrt(2) in the y direction and 2 in the x direction. **