IDZ Ryabushko 4.2 Option 22

No. 1. It is necessary to construct surfaces and determine their type for the following equations: a) x2 = 5(y2 + z2); b) 2x2 + 3y2 – z2 = 36.

No. 2. For the given equations, it is necessary to write down the equation of the surface obtained by rotating this line around the specified coordinate axis and make a drawing: a) y2 = 5z; Oz; b) 3x2 + 7y2 = 21; Ox.

No. 3. It is necessary to construct a body limited by the indicated surfaces: a) z = 16x2 + y2; z = 0; y = 2x; y = 0; x = 1. b) z – 4 = 6(x2 + y2); z = 4x + 1.

Let's move on to solving problems:

No. 1. a) The equation x2 = 5(y2 + z2) describes a two-sheet hyperboloid, the axes of which are directed along the y and z axes. b) The equation 2x2 + 3y2 – z2 = 36 defines the surface of the ellipsoid.

No. 2. a) The equation y2 = 5z, when rotated around the Oz axis, generates the surface of a cone. Figure: b) The equation 3x2 + 7y2 = 21, when rotated around the Ox axis, generates the surface of an ellipsoid. Drawing:

No. 3. a) Given a bounded body bounded by the surfaces z = 16x2 + y2, z = 0, y = 2x, y = 0 and x = 1. The first two equations define a parabolic paraboloid parallel to the xz plane, and y = 2x and y = 0 define planes parallel to the yz plane. x = 1 specifies the vertical plane. Thus, the limited body has the shape of a truncated pyramidal column. b) A bounded body is given, bounded by the surfaces z – 4 = 6(x2 + y2) and z = 4x + 1. The first equation defines an elliptic paraboloid with a vertex at the point (0, 0, 4) and semi-axes directed along the x and y axes . The second equation specifies a plane parallel to the yz plane. Thus, the bounded body has the shape of a truncated cone, the vertex of which is located at the point (0, 0, 4).

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IDZ Ryabushko 4.2 Option 22 is a digital product containing tasks for independent work in mathematics, developed based on the textbook by the author V.F. Ryabushko. In this product you will find tasks with detailed instructions and answers to problems from various areas of mathematics.

In particular, this product contains tasks on constructing surfaces and determining their type, recording equations of surfaces obtained by rotating lines around coordinate axes, as well as on constructing bodies bounded by given surfaces.

By purchasing "IDZ Ryabushko 4.2 Option 22" in our store, you receive useful material for self-preparation for exams and olympiads in mathematics. Our store guarantees fast and convenient product delivery, as well as 24-hour technical support.

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IDZ Ryabushko 4.2 Option 22 is a task for students who study mathematics and geometry. The assignment contains several problems that need to be solved. In the first task it is necessary to construct surfaces and determine their appearance. In the second task, you need to write down an equation and determine the type of surface obtained by rotating a given line around a specified coordinate axis, and also draw it. The third problem requires constructing a body bounded by the specified surfaces.

In the first problem the surface equations are given: a) x2 = 5(y2 + z2); b) 2x2 + 3y2 – z2 = 36. It is necessary to construct these surfaces and determine their type.

In the second problem, you need to construct surfaces obtained by rotating the lines: a) y2 = 5z around the Oz axis; b) 3x2 + 7y2 = 21 around the Ox axis. It is required to write down the equation of the surface and determine its type, as well as draw the resulting surface.

In the third problem, you need to construct a body bounded by surfaces: a) z = 16x2 + y2; z = 0; y = 2x; y = 0; x = 1; b) z – 4 = 6(x2 + y2); z = 4x + 1. You need to draw a body and determine its volume.

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