Name Date MULTIVARIABLE CALCULUS FINAL PROJECT Summary: A sliceform is a three-dimensional shape assembled from flat slices that join together to form a grid structure. After researching sliceforms, your task is to create a template for a sliceform that represents a three-dimensional model that we have worked with this year (such as a cone, tetrahedron, hyperboloid, torus, hyperbolic paraboloid, ellipsoid, etc.). You will use this template to create a three- dimensional model of your sliceform and calculate its volume, mass, center of gravity, and centroid. For your convenience, given a three-dimensional solid G and a continuous density function S(x, y, z) : Mass of G: Center of Gravity (x, y, z) : Centroid (x, y, z) : M = [ S(x, y, z) dV x = fix . 8(x, y, z) dV x= _ fff xav G y =fff x . 8(x, y,z) dV y = SIS yay z= ]][ x . 8(x, y,z ) dV z = SIS zdv Requirements: You must... select a sliceform that no one else in the class has selected (this means, as a class, you will need to coordinate which student creates which model - use the Google Doc posted in Google Classroom to do this). locate and save a model picture of the solid you wish to recreate (for reference purposes). . create the sliceform template (do not make the template too small or it will be difficult to work with) define the three dimensional solid, G, you will use . write an equation for the density function, o(x, y, z) , you will use build your three-dimensional model of G calculate its volume calculate its mass locate the center of gravity locate the centroidFor the cone: Definition of Solid G: The solid G is a cone with a circular base and a curved surface that tapers to a point (apex). It has one curved face and one circular base. E uation for Densit Function 2: To calculate the mass and other properties of the cone, we need a density function. Let's assume a constant density for simplicity: x. y. 2) = 9 Volume of G: The volume of a cone can be calculated using the formula: Volume = (1/3} * rt * r"2 * h, where r is the radius of the base and h is the height of the cone. Mass of G: The mass of the cone can be calculated using the formula: Mass = Volume * Density, substituting the density function, Mass =(1/3}* rt * r"2 * h * p. Center of Gravity (x, y, z): The center of gravity of the cone can be calculated using the formulas: X=(1/M)*IIIX*(X.V.Z)51M, y= (1/Ml*fify*f_zlx.y, ztsili, z = (1/M} * III 2 * got, y, 2) SIM: where M is the mass of the cone. Centroid (x, y, z): The centroid coordinates of the cone can be obtained using the formulas: X= (1m*IIIX*xlysz)gJ y= (1N}*my*x.y, 219M, 2 = (1N) * H] z * x, y, Ill-1!, where V is the volume of the cone