7 on paper please **Problem 7:****Objective:** Sketch the surface defined by the inequality:\[ 1 \leq z \leq 2 - x^2 - y^2 \]**Explanation for the inequality:**The given inequality describes a region in three-dimensional space (ℝ³).1. **Lower Bound:** \( z \geq 1 \) - This represents a horizontal plane at \( z = 1 \). 2. **Upper Bound:** \( z \leq 2 - x^2 - y^2 \) - This is the equation of an upward-opening paraboloid, which opens downwards because of the minus sign before \( x^2 \) and \( y^2 \).**Intersection of the bounds:**- When \( z = 2 - x^2 - y^2 \), substituting \( z = 1 \): \[ 1 = 2 - x^2 - y^2 \] \[ x^2 + y^2 = 1 \]- This represents a circle in the \( x \)-\( y \) plane with a radius of 1 centered at (0,0).**Three-Dimensional Shape Description:**- The surfaces to be sketched are between the plane \( z = 1 \) and the paraboloid \( z = 2 - x^2 - y^2 \).- The region of interest lies within the circular boundary where \( x^2 + y^2 \leq 1 \) and between \( z = 1 \) and the paraboloid.**Graph's Characteristics and Drawing:**- **Base:** Circle in the \( x \)-\( y \) plane with radius 1.- **Curved Upper Surface:** A portion of the paraboloid capped at \( z = 1 \).The overall region is a "cap" of the paraboloid cut off by the plane \( z = 1 \).To accurately sketch this surface:1. Draw the circle \( x^2 + y^2 \leq 1 \) in the \( x \)-\( y \) plane.2. Represent the plane \( z = 1 \) above this base circle.3. Show the paraboloid \( z = 2 - x^2 - y^2 \) with the cap created by the intersection at \( z =

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Answered: 7. Sketch the surface 1≤z≤2-x² - y²

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7. Sketch the surface 1≤z≤2-x² - y²

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter12: Conic Sections

Section12.1: Parabolas

Problem 1E: A parabola is the set of all points in the plane that are equidistant from a fixed point called the...

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