Symmetry What symmetry will you find in a surface that has an equation of the form r = f(z) in cylindrical coordinates? Give reasons for your answer.
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A: Hey, since there are multiple questions posted, we will answer first question. If you want any…
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A: For t=2, x=t2+1=22+1=4+1=5For t=2, y=5-t3=5-23=5-8=-3
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Q: Parametric equations and a value for the parameter t are given x = (80cos 45°)t, y = 6 + (80sin…
A: For t=2, x = (80cos 45°)t = (80cos 45°)(2)= 80×22×2=802For t=2, y = 6 + (80sin 45°)t - 16t2= 6 +…
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A: Given That - The cylindrical equation, r = 6
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Q: Symmetry What symmetry will you find in a surface that hasan equation of the form r = ƒ(f) in…
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A: Given:
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- What symmetry will you find in a surface that has an equation of the form r = ƒ(z) in cylindrical coordinates? Give reasons for your answer.a. Show that when you express ds2 = dx2 + dy2 + dz2 in terms of cylindrical coordinates, you get ds2 = dr2 + r2 du2 + dz2. b. Interpret this result geometrically in terms of the edges and a diagonal of a box. Sketch the box. c. Use the result in part (a) to find the length of the curve r = eu, z = eu, 0 <=u<=ln 8.Minimizing polar inertia A thin plate of constant density is tooccupy the triangular region in the first quadrant of the xy-planehaving vertices (0, 0), (a, 0), and (a, 1 > a). What value of a willminimize the plate’s polar moment of inertia about the origin?
- Moment of inertia of solid cone Find the moment of inertia ofa right circular cone of base radius a and height h about its axis.(Hint: Place the cone with its vertex at the origin and its axisalong the z-axis.)The line segment from (1, −1, 3) to (0, −3, 1) forms a diameter of the sphere S1. Find the center-radius form of the equation of S1. Consider the parabola E : x = z2 − 4 in the xz-plane. Give an equation for the surface of revolution generated when E is revolved about the x-axis. Sketch the portion of the graph of the cylinder whose equation is the same as E in between the xz-plane and the plane y = 4. Label all important points such as intercepts, vertices of parabolas and hyperbolas, endpoints of the major and minor axes of ellipses, and end- points of traces.(a) Find parametric equations for the portion of the cylinder x² + y² = 5 that extends between the planes z = 0 and z = 1. (b) Find parametric equations for the portion of the cylinder x² + z² = 4 that extends between the planes y = 1 and y = 3.
- The plane y + z = 2 intersects the cylinder x2 + y2 = 5 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (2, 1, 1). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)Use spherical coordinates to describe the region above the xy-plane between the spheres of radius 1 and 3 centered at the origin. Determine the Cartesian equation of the surface with spherical coordinate equation ρ = 2cosθsinφ−2sinθsinφ+2cosφ. It turns out this describes a sphere. What is the center and radius of this sphere?Find an equation in rectangular coordinates for the surface represented by the cylindrical equation. r = 6 Sketch its graph
- Set up only. The base of a solid is an elliptical region with the boundary curve 9 x 2 + 4 y 2 = 36. Cross- sections are perpendicular to the x-axis are isosceles right triangles with one leg on the base.Surface Area for Implicit and Explicit Forms Find the area of the ellipse cut from the plane z = cx (c a constant)by the cylinder x2 + y2 = 1.Parametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cylinder y2 + z2 = 36, for 0 ≤ x ≤ 9