IDZ Ryabushko 4.2 Option 10

No. 1. It is necessary to construct surfaces and determine their type for the equations: a) 5z2 + 2y2 = 10x; b) 4x2 – 3y2 – 5z2 + 60 = 0.

a) The equation 5z2 + 2y2 = 10x defines a parabolic cylinder with a vertical axis. The axis of the cylinder is the x axis, and the vertices of the parabolas belong to the yz plane.

b) The equation 4x2 – 3y2 – 5z2 + 60 = 0 defines an elliptic paraboloid. The axis of a paraboloid is the x-axis, and the directions of the parabola are the y and z axes.

No. 2. It is necessary to write down the equation and determine the type of surface obtained by rotating this line around the specified coordinate axis and draw a picture for: a) y2 – 5x2 = 5; Oy; b) y = 3; z = 1; Ox.

a) The equation y2 – 5x2 = 5 defines a hyperbola with its center at the origin and axes parallel to the coordinate axes. When rotated around the Oy axis, a hyperbolic paraboloid is obtained.

b) The equation y = 3 defines a plane parallel to the Oxz axes. When rotated around the Ox axis, a cylinder is obtained.

The equation z = 1 defines a plane parallel to the Ox and Oy axes. When rotating around the Ox axis, a cone is obtained.

No. 3. It is necessary to construct a body limited by the indicated surfaces: a) y = 4x; y = 0; x = 1; ... ; z = 0.

Let's look at the surfaces one by one:

• y = 4x specifies a plane passing through the origin and making a 45 degree angle with the xy plane.
• y = 0 specifies a plane parallel to the xy plane and passing through the z axis.
• x = 1 specifies a plane perpendicular to the xy plane and passing through the point (1, 0, 0).
• z = 0 specifies a plane parallel to the xy plane and passing through the origin.

Thus, the body is limited by the planes y = 4x, y = 0, x = 1 and z = 0. This body is a triangular prism with a height equal to 1.

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IDZ Ryabushko 4.2 Option 10 is a task for constructing surfaces, determining their type, writing equations of surfaces and constructing a body bounded by given surfaces.

Task No. 1 requires constructing surfaces and determining their form for the equations 5z2 + 2y2 = 10x and 4x2 – 3y2 – 5z2 + 60 = 0.

Task No. 2 requires writing down an equation and determining the type of surface obtained by rotating a given line around a specified coordinate axis and making a drawing. For task a) the equation of the line y2 – 5x2 = 5 rotates around the Oy axis, and for task b) the line y = 3 rotates around the z = 1 axis, perpendicular to the Ox axis.

Task No. 3 requires constructing a body bounded by surfaces y = 4x, y = 0, x = 1 and z = 0.

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