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- find an equation of the form r = f (θ , z) in cylindrical coordinates for the following surfaces. x2 + y2 = 4What is the size of the largest segment parallel to the y axis inside the ellipsoid determined by the equation below? 4x2+5y2+3z2+6x+8y+5z−4=0Find an equation for the paraboloid z=x^2+y^2 in spherical coordinates.
- Find the center C and the radius a for the spheres x2 + y2 + z2 + 4x - 4z = 0Suppose that a cylindrical container of radius r and height L is filled with a liquid with volume V , and rotated along the y-axis with constant angular speed ω. This makes the liquid rotate, and eventually at the same angular speed as the container. The surface of the liquid becomes convex as the centrifugal force on the liquid increases with the distance from the axis of the container. The surface of the liquid is a paraboloid of revolution generated by rotating the parabola y = h + ω2x2/2g around the y-axis, where g is gravitational acceleration and h is shown below. (You can take g=32ft/s2 or 9.8m/s2). Express h as a function of ω. (2) At what angular speed ω will the surface of the liquid touch the bottom? At what speed will it spill over the top? (3) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed ω. The surface of the liquid is 5 ft below the top of the tank at the central…Find the center C and the radius a for the spheres (x - 1)2 + (y - 2)2 + (z + 1)2 = 103 + 2x + 4y - 2z
- Find an equation in rectangular coordinates for the surface represented by the cylindrical equation r2 cos 2 + z2 + 1 = 0Sketch the solid that has the given description in spherical coordinates. 0 ≤ θ ≤ π 0 ≤ ϕ ≤ π/2 1 ≤ p ≤ 3Consider the ellipse E in the xy-plane defined by the equation ax2 + y2 = 1 where a is positive. (1) Find a parametrization r(t) of E (2) Find all the points where r(t) is orthogonal to r'(t).
- find an equation of the form ρ = f (θ , φ) in spherical coordinates for the following surfaces x2 − y2 = 4Consider the ellipsoid 3x2+2y2+z2=12. The implicit form of the tangent plane to this ellipsoid at (1,2,-1) is ? The parametric form of the line through this point that is perpendicular to that tangent plane is L(t)=?Try to sketch by hand the curve of intersection of the parabolic cylinder y = x2 and the top half of the ellipsoid x2 + 5y2 + 5z2 = 25. Then find parametric equations for this curve. (x(t), y(t), z(t)) = for −1.5 ≤ t ≤ 1.5