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- Find a parametrization of the hyper-boloid of two sheets (z2/c2)-(x2/a2)-(y2/b2)=1The upper half of the unit sphere x2+y2+z2=1 is z=Square root of 1-x2-y2. Find its centroid?An artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it on display. The artwork has constant density and must be balanced at its centroid. The shape of the artwork was created on a computer program then casted and fabricated. The following equation was put into the computer to generate the shape:y=2sin(πx)+5y=2sin(πx)+5 bounded by x = 0, x = 2, and y = 0Draw the Lamina in an x-y plane and put a dot where the centroid should be. Show all work and formulas you are using.The centroid is at (¯x,¯y) where ¯x = ¯y
- A thin plate of constant density is to occupy the triangular region in the first quadrant of the xy-plane having vertices (0, 0), (a, 0), and (a, 1/a). What value of a will minimize the plate’s polar moment of inertia about the origin?An artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it on display. The artwork has constant density and must be balanced at its centroid. The shape of the artwork was created on a computer program then casted and fabricated. The following equation was put into the computer to generate the shape:y=2sin(πx)+5 bounded by x = 0, x = 2, and y = 0Draw the Lamina in an x-y plane and put a dot where the centroid should be. Show all work and formulas you are using.The centroid is at (¯x,¯y) whereAn artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it on display. The artwork has constant density and must be balanced at its centroid. The shape of the artwork was created on a computer program then casted and fabricated. The following equation was put into the computer to generate the shape: y=1sin(πx)+5 bounded by x = 0, x = 2, and y = 0 Draw the Lamina in an x-y plane and put a dot where the centroid should be. Show all work and formulas you are using. The centroid is at (¯x,¯y), where ¯x= ¯y=
- Find the points on the sphere at which the minimum and maximum distances between the sphere x2+y2+z2= 58 and the point (2,3,4) occur.Find the Largest open rectangle in the plane in which the hypotheses of Existence and uniqueness Theorem are satisfied fory'=−2t/(1 + y^3), y(1) = 1 Describe then sketch the regions.5.Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0.Vx,y,z=rx2+σy2+σz−2r2=c>0. a.Calculate dVdtdVdt along trajectories of the Lorenz equations (1).