IDZ Ryabushko 4.2 Option 8

Construct surfaces and determine their type: a) 4x² – 5y² – 5z² + 40 = 0; Solution: This equation defines a second-order surface, which is called a one-sheet hyperboloid. It has two intersecting real surfaces that look like two connected hyperbolic baths.

б) y = 5x² +3z². Solution: This equation defines a parabolic cylindrical surface oriented along the y-axis. It has an infinite number of pairwise parallel cross sections, which are parabolas.

Write down the equation and determine the type of surface obtained by rotating this line around the indicated coordinate axis: a) 3x² – 5z² = 15; Oz; Solution: To obtain a surface, it is necessary to rotate this line around the Oz axis. The resulting surface is a hyperbolic paraboloid with the vertex located on the Oz axis.

b) z = – 1; y = 3; Ox. Solution: To obtain a surface, it is necessary to rotate this line around the Ox axis. The resulting surface is a conical paraboloid with the vertex located on the Ox axis.

Construct a body bounded by the indicated surfaces: a) y = 2x; y = 0; x = 2; z = xy; z = 0. Solution: This body is bounded by two planes (y = 0 and x = 2) and two surfaces (y = 2x and z = xy). It is a pyramid with a triangular base.

b) x² + y² = z²; x² + y² = 1; y ≥ 0; z ≥ 0. Solution: This body is bounded by two parabolic cylinders (x² + y² = z² and x² + y² = 1), plane y = 0 and plane z = 0. It is a truncated cone with a circular base, bounded by plane y = 0 below and plane z = 0 above.

IDZ Ryabushko 4.2 Option 8

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IDZ Ryabushko 4.2 Option 8 is a digital product containing solutions to problems in mathematics from Individual homework compiled by Ryabushko. The IPD includes problems in algebra, geometry, trigonometry and mathematical analysis. Each problem is accompanied by a detailed solution that will help students better understand the material and prepare for the exam.

Problem No. 1 requires constructing surfaces and determining their appearance. The first surface is given by the equation 4x2 – 5y2 – 5z2 + 40 = 0 and is a one-sheet hyperboloid. The second surface is given by the equation y = 5x2 + 3z2 and is a parabolic cylindrical surface oriented along the y-axis.

In task No. 2, it is necessary to write down an equation and determine the type of surface obtained by rotating a given line around the specified coordinate axis, and also construct a drawing. In the first case, a line 3x2 – 5z2 = 15 is given, which must be rotated around the Oz axis. The resulting surface is a hyperbolic paraboloid with the vertex located on the Oz axis. In the second case, the line z = – 1 is given; y = 3, which must be rotated around the Ox axis. The resulting surface is a conical paraboloid with the vertex located on the Ox axis.

Problem No. 3 requires constructing a body bounded by the specified surfaces. In the first case, the body is limited by two planes (y = 0 and x = 2) and two surfaces (y = 2x and z = xy) and is a pyramid with a triangular base. In the second case, the body is bounded by two parabolic cylinders (x2 + y2 = z2 and x2 + y2 = 1), the plane y = 0 and the plane z = 0 and is a truncated cone with a circular base, bounded by the plane y = 0 below and the plane z = 0 on top.

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IDZ Ryabushko 4.2 Option 8 is a set of problems in mathematical analysis and analytical geometry. In the tasks of this option, you need to construct surfaces, write down the equations of the surfaces, determine their type, and also construct bodies bounded by the specified surfaces.

In particular, in problem No. 1 it is required to construct surfaces and determine their form for the equations 4x2 – 5y2 – 5z2 + 40 = 0 and y = 5x2 + 3z2. In problem No. 2, it is necessary to write down equations and determine the type of surfaces obtained by rotating a given line around the indicated coordinate axes. Problem No. 3 requires constructing bodies bounded by the specified surfaces for the equations y = 2x, y = 0, x = 2, z = xy, z = 0 and x2 + y2 = z2, x2 + y2 = 1, y ≥ 0, z ≥ 0.

This set of tasks is intended to consolidate and deepen knowledge of analytical geometry and mathematical analysis, as well as to develop skills in constructing surfaces and bodies in space.

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