Question 1

Which of the following equations represents an elliptic paraboloid?

Accepted Answer

$$z = x^2$ + y^2$

Question 2

A surface defined by $$x^2$$ + $$y^2 = 1$ is best described as:

Accepted Answer

A cylinder parallel to the $$z-$$axis

Question 3

Which quadric surface has horizontal traces that are ellipses and vertical traces that are hyperbolas?

Accepted Answer

Hyperboloid of one sheet

Question 4

Given the equation $$x^2$$ + $$y^2$$ + $$z^2 = 9$, what is the geometric shape of the surface?

Accepted Answer

Sphere with radius 3

Question 5

Which of the following is the standard form of a quadric surface equation?

Accepted Answer

$$Ax^2$$ + $$By^2$$ + $$Cz^2 + $$Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$

Question 6

If the equation of a surface is $$y^2$$ + $$z^2 = 4$, which variable is missing and what does this imply about the surface?

Accepted Answer

The variable $$x is $$missing; the surface is a cylinder parallel to the $$x-$$axis.

Question 7

Classify the surface given by $$4x^2$$ - $$y^2$$ + $$2z^2 + 4 = 0$.

Accepted Answer

Hyperboloid of two sheets

Question 8

Which surface is described by the equation $$z^2$$ = $$x^2$$ + $$y^2$$?

Accepted Answer

Cone

Question 9

A medical imaging device scans a region described by $$\frac{x^2$}{9} + \frac{y^2$}{4} + \frac{z^2$}{16} = 1. $$What is the shape of the scanned region?

Accepted Answer

Ellipsoid

Question 10

Which of the following equations represents a hyperboloid of one sheet?

Accepted Answer

$$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = 1$$

Question 11

A surface is defined by $$z = x^2$ - y^2$. What is the name of this surface?

Accepted Answer

Hyperbolic paraboloid

Question 12

Which quadric surface is used to model the shape of a satellite dish, given by $$z = x^2$ + y^2$?

Accepted Answer

Elliptic paraboloid

Question 13

If a surface is defined by $$x^2$$ + $$y^2$$ - $$z^2 = 1$, what is the type of this surface?

Accepted Answer

Hyperboloid of one sheet

Question 14

Which of the following surfaces has traces in the $$x$$-$$y$$ plane that are ellipses and traces in the $$x$$-$$z$$ and $$y$$-$$z$$ planes that are parabolas?

Accepted Answer

Elliptic paraboloid

Question 15

A surface is described by $$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = 1. $$What is the name of this surface?

Accepted Answer

Hyperboloid of two sheets

Question 16

Which surface is described by the equation $$z = x^2$ + y^2$ and is commonly used in the design of reflectors and antennas?

Accepted Answer

Elliptic paraboloid

Question 17

A surface is defined by $$x^2$$ + $$y^2 = 4$. What is the geometric interpretation of this surface?

Accepted Answer

A circular cylinder parallel to the $$z-$$axis

Question 18

Which of the following equations represents a cone?

Accepted Answer

$$z^2$$ = $$x^2$$ + $$y^2$$

Question 19

A surface is defined by $$z = x^2$ - y^2$. What is the shape of the cross-section in the x$-z$ plane?

Accepted Answer

Parabola

Question 20

Which quadric surface is described by the equation $$x^2$$ + $$y^2$$ + $$z^2$$ = $$r^2$$?

Accepted Answer

Sphere

Section 2.6 - Cylinders and Quadric Surfaces

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