Question

Topic: Quadric surfaces.

I have two homework problems, and they are quite a doozy for me. So the point is, I am given details of a quadric surface on the Cartesian coordinate system, and thus have to use the data to construct an equation through it.

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Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is 4 times the distance from P to the yz-plane.

(1) So this a cone, so its equation will be the form of (x^2/a^2)+(y^2/b^2)-(z^2/c^2)=0.

Unfortunately, this is about as far as I can get. Same situation with the next one.

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A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 280 m and the minimum diameter, 500 m above the base, is 180 m. Find an equation for the tower. (Assume the center is at the origin with axis the z-axis and the minimum diameter is at the center.)

(1) A hyperboloid of one sheet with horizontal circles will have the equation of (x^2/a^2)+(y^2/a^2)-(z^2/b^2)=1.

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Honestly, the only attempts I have made have been on the second problem. My strategy was at first to set z=0 and find that the horizontal traces should construct a circle with radius of 180 m. But now I get stuck -- don't hyperboloids of one sheet extend infinitely? How do I know if the height is 1,000 m? I'm pretty sure I don't have to worry about the bases extending to a radius of 280 m since those will extend with height, but determining the height is a different animal to me.

The first problem, I am a little lost in the language.

Insight is appreciated. I do not expect you to hand me a silver platter answer.

Thanks.

Answer 1

For P:(x,y,z)
distance from P to the x-axis: √(y^2+z^2)
distance from P to the yz-plane: x
So,
√(y^2+z^2) = 4x
y^2+z^2 = 16x^2

x^2/1 - y^2/16 - z^2/16 = 0

I'll get back to you on the rest.