This worksheet will challenge students’ understanding of volume and the equation l x w x h = V. Then, once they solve the equation have them draw the 3-D objects. The objects will all be rectangular prisms. Name Figure Equation Hyperbolic Cylinder Ellipsoid Hyperbolic Paraboloid Hyperboloid of One Sheet Elliptic Cylinder Cone Elliptic Paraboloid Parabolic Cylinder Hyperboloid of Two Sheets The Surface Plots: The Equations: a. z2 + y2 − x2 = 0 b. x2 − y2 + z = 0 c. 1 2 y2 − x2 − 2z2 = 0.1 d. z2 + 2y2 = 5 e. x2 + 2z = 0 f. x2 + 1 2 z2 − y = 0 g. x2 + 1 4 y2 + 1 2 z2 = 1 h. z2 − x2 = 1 Name Figure Equation Hyperbolic Cylinder Ellipsoid Hyperbolic Paraboloid Hyperboloid of One Sheet Elliptic Cylinder Cone Elliptic Paraboloid Parabolic Cylinder Hyperboloid of Two Sheets The Surface Plots: The Equations: a. z2 + y2 − x2 = 0 b. x2 − y2 + z = 0 c. 1 2 y2 − x2 − 2z2 = 0.1 d. z2 + 2y2 = 5 e. x2 + 2z = 0 f. x2 + 1 2 z2 − y = 0 g. x2 + 1 4 y2 + 1 2 z2 = 1 h. z2 − x2 = 1 A quadric surface given by an equation of the form (x 2 / a 2 ) ± (y 2 / b 2 ) - (z 2 / c 2 ) = 1; in certain cases it is a hyperboloid of revolution, which can be realized by rotating the pieces of a hyperbola about an appropriate axis.

Comparing (A.17) with the equations for the hyperboloids of one and two sheet we see that the cone is some kind of limiting case when instead of having a negative or a positive number on the l.h.s. of the quadratic equation we have exactly 0. Start studying Quadric Surfaces (Calculus III). Learn vocabulary, terms, and more with flashcards, games, and other study tools. ... hyperboloid of one sheet (1 ... The two-sheeted hyperboloid is the only non-connected quadric. The two-sheeted hyperboloid of revolution can be defined as the surface of revolution generated by the rotation of a hyperbola around its transverse axis. It is the locus of the points M satisfying , where F and F' are the common foci of these hyperbolas. of the hyperboloid z = x2 2y2 and the cylinder x2 +y = 1. Solution: There are several correct solutions–here is one. We can parameterize motion on the circle at x(t) = cos(t), y(t) = sin(t) (2 points) Next, we use the equation of the hyperboloid to conclude that z = cos2(t) sin2(t) (2 points) Putting all the pieces together, we conclude that 2016 HigHer ScHool certificate examination REFERENCE SHEET – Mathematics – – Mathematics Extension 1 – – Mathematics Extension 2 – trization of the hyperboloid of one sheet on page 313, but this has the disad-vantage of not showing the rulings. Let us show that the hyperboloid of one sheet is a doubly-ruled surface by finding two ruled patches on it. This can be done by fixing a,b,c > 0 and defining x±(u,v) = α(u) ±vα0(u)+v(0,0,c), where

of the hyperboloid z = x2 2y2 and the cylinder x2 +y = 1. Solution: There are several correct solutions–here is one. We can parameterize motion on the circle at x(t) = cos(t), y(t) = sin(t) (2 points) Next, we use the equation of the hyperboloid to conclude that z = cos2(t) sin2(t) (2 points) Putting all the pieces together, we conclude that Define hyperboloid. hyperboloid synonyms, hyperboloid pronunciation, hyperboloid translation, English dictionary definition of hyperboloid. hyperboloid top: hyperboloid of one sheet bottom: hyperboloid of two sheets n. Volume of a Hyperboloid of One Sheet A hyperboloid of one sheet is the surface obtained by revolving a hyperbola around its minor axis. Denote the solid bounded by the surface and two planes \(y=\pm h\) by \(H\). 7. (22 points) Sketch (or describe) the solid whose volume is given by a. 2 242 0 0 0 yy dxdzdy ³ ³ ³ b. 4 2 4 00r rdzd dr S ³³³ T c. 2 /4 3 2 0 0 0 sin SS ³ ³ ³U I U I Td d d 8. (12 points)Find the volume of the solid inside the sphere x y z2 2 2 9 and inside the cylinder xy22 1. Two masses, m 1 and m 2, separated by a distance, r, attract each other with a gravitational force, given by the following equations, in proportion to the gravitational constant G: Moments of inertia The rotational equivalent of mass is inertia, I , which depends on how an object’s mass is distributed through space.

For a vector normal to the plane we can choose the cross product of two vectors in the plane:4( 1) 3( 2) 4( 3) 0 4 3 4 is normal to the plane 4 3 4 10n i j kx y z8 6 8i j kx y z . and 2, 4, 1v= PR . Two distinct planes in 3-space either are parallel or intersect in a line. (a) Hyperboloid of two sheets (c) Hyperboloid of one sheet (e) None of these (d) Elliptic cone 7. Find the equation of revolution if the generating curve, y-2 1, is revolved about t he x-axis. If k is positive, say k = 1 we get a surface that is called a hyperboloid of one sheet. Let's assume that a, b and c are all 1. These constants only scale the surface in the x, y and z directions anyway. What we have now is the function x 2 + y 2 - z 2 = 1 or x 2 + y 2 = z 2 + 1. The contours of this surface are circles.

5.) a set of parametric equations for the line through (1, 2, 3) with v as its direction vector . 6.) For each of the graphs below, select an equation from the list of equations, that could give the graph. Also choose the name of the surface from the list of names. Use the appropriate letter to indicate your choice. A simple lathe for making hyperboloid lenses has been developed. A cutting tool revolves around the vertical tool spindle, and its edge forms a hyperboloid of one sheet of revolution. A plastic rod is chucked on the main spindle and set at right angles to the hyperboloid of one sheet of revolution formed by the edge of the cutting tool. These free volume of a cylinder worksheets will help your students practice calculating the volume of a cylinder by using a formula. This set of problems features a graphic of a cylinder and the length of the radius of the cylinder is given. If k is positive, say k = 1 we get a surface that is called a hyperboloid of one sheet. Let's assume that a, b and c are all 1. These constants only scale the surface in the x, y and z directions anyway. What we have now is the function x 2 + y 2 - z 2 = 1 or x 2 + y 2 = z 2 + 1. The contours of this surface are circles. In the second case (−1 in the right-hand side of the equation), one has a two-sheet hyperboloid, also called elliptic hyperboloid. The surface has two connected components, and a positive Gaussian curvature at every point.

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ellipsoid hyperboloid of 1 sheet hyperboloid of 2 sheets paraboloid something else The following problem is optional. You may earn extra points, but work on this Hyperboloid of two sheets synonyms, Hyperboloid of two sheets pronunciation, Hyperboloid of two sheets translation, English dictionary definition of Hyperboloid of two sheets. hyperboloid top: hyperboloid of one sheet bottom: hyperboloid of two sheets n. Math 259 Winter 2009 Recitation Handout 5: Sketching 3D Surfaces If you are simply given an equation of the form z = f(x, y) and asked to draw the graph, producing the graph can be a very complicated and difficult operation even for relatively simple functions. On the total volume of the double hyperbolic space: ... contain both the two sheets of the two-sheeted hyperboloid (but with the geodesics in the lower sheet $\mathbb ... Describe the traces of the hyperboloid of one sheet given by equation x 2 3 2 + y 2 2 2 − z 2 5 2 = 1. x 2 3 2 + y 2 2 2 − z 2 5 2 = 1. Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure 2.85 (a).

Hyperboloid two sheet equation for volume

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The image shows a one-sheeted hyperboloid symmetric around the axis. The blue curve is the unique hyperboloid geodesic passing through the given point (shown in black) and intersecting the parallel (i.e. the circle of latitude) through that point at the given angle . t2 1sin ct 3 5 The hyperboloid of one sheet is a doubly ruled surface. Through each its points there are two lines that lie on the surface. Both kinds of circular hyperboloids as well as the cone can be included in one family of surfaces by modifying their de ning equations slightly. Consider the equations x2 + y 2= z + e where eis a constant. A quadric surface given by an equation of the form (x 2 / a 2 ) ± (y 2 / b 2 ) - (z 2 / c 2 ) = 1; in certain cases it is a hyperboloid of revolution, which can be realized by rotating the pieces of a hyperbola about an appropriate axis. For a vector normal to the plane we can choose the cross product of two vectors in the plane:4( 1) 3( 2) 4( 3) 0 4 3 4 is normal to the plane 4 3 4 10n i j kx y z8 6 8i j kx y z . and 2, 4, 1v= PR . Two distinct planes in 3-space either are parallel or intersect in a line. 7. (22 points) Sketch (or describe) the solid whose volume is given by a. 2 242 0 0 0 yy dxdzdy ³ ³ ³ b. 4 2 4 00r rdzd dr S ³³³ T c. 2 /4 3 2 0 0 0 sin SS ³ ³ ³U I U I Td d d 8. (12 points)Find the volume of the solid inside the sphere x y z2 2 2 9 and inside the cylinder xy22 1. Vrhdy if slices are horizontal Physical Applications: Physics Formulas Associated Calculus Problems Mass: Mass = Density * Volume (for 3‐D objects) Mass = Density * Area (for 2‐D objects) Mass = Density * Length (for 1‐D objects) Mass of a one‐dimensional object with variable linear density: () ().