Brandon TomekCEO Regal Goblets Inc.Math 152-20224 April 2014Instructor: Dr. Philip YasskinGoblet Design"Le Gobelet Royal"1 unit = 1 cmrestart;Packages needed for plotting and analysiswith(plots):with(Student[Calculus1]):________________________________________________________________________Equations of sectional lines for exterior shape of gobletEquation for the basebase:=x->3.25-3.75*x^2;Equation for the stemstem:=x->0.5+.045*sin(2*Pi*x);Equation for the stem bulbstembulb:=x->-(x-4.75)^(2)+1.30;Equation for the cupcup:=x->a*x^3+b*x^2+c*x+d;______________________________________________________________________________Equations equating different sections of the goblet with respect to the cup equationEquating the cup and stem equation at x=7.5eqA:=cup(7.5)=stem(7.5);Setting the cup's radius equal to 3.85 at x=11eqB:=cup(11)=3.85;Setting the cup's radius equal to 3.5 at x=12.1eqC:=cup(12.1)=3.5;Setting the cup's radius equal to 3.425 at x=15.4eqD:=cup(15.4)=3.425;_____________________________________________________________________________Solving for the values of a, b, c, and d that satisfy the above criteria and assigning them tovalues:=solve({eqA,eqB,eqC,eqD},{a,b,c,d});assign(values);________________________________________________________________________________Exterior Goblet ShapePiecewise function assigning equations to different sections of the shape A:=x->piecewise(x<.75,base(x),x>.75 and x<4.1,stem(x),x>4.1 and x<5.25,stembulb(x),x>5.25 and x<7.5,stem(x),x>=7.5,cup(x));Plotting the exterior goblet shapeplot(A(x),x=0..15.4,y=0..8.8,scaling=constrained);__________________________________________________________________________________Interior Cup SurfaceEquation for the line bounding the inside of the cup width(x):=cup(x)-0.35;Piecewise function assigning the width equation to the cup sectionB:=x->piecewise(x<=4.4,0,x>7.5,width(x));Plotting the width equationplot(B(x),x=0..15.4,y=0..8.8,scaling=constrained);plot([A(x),B(x)],x=0..15.4,y=0..8.8,scaling=constrained);___________________________________________________________________________________________Calculation and EvaluationRotate exterior shape and interior surface to obtain a 3-D volume of revolutionVolumeOfRevolution(A(x),B(x),x=0..15.4,output=plot,orientation=[0,165],title='Goblet',scaling=constrained, axes=none);VolumeOfRevolution(A(x),B(x),x=0..15.4,output=plot,orientation=[0,165],title='Goblet',scaling=constrained, axes=none, caption="");This cannot 3D print because it is missing the surfaces to close off the bottom and the top rim. - PYEvaluate the capacity of the cup by finding the volume of revolution of the interior surface bounded by the x-axisCupCapacity:=evalf(VolumeOfRevolution(B(x),0,x=0..15.4,output=integral));Evaluate the volume of the goblet by finding the volume of revolution of the outer shape bounded by the interior surfaceGlassVolume:=evalf(VolumeOfRevolution(A(x),B(x),x=0..15.4,output=integral));Evaluate the center of mass CenterOfMass:=evalf(int(x*((A(x))^2-(B(x))^2),x=0..15.4)/int((A(x))^2-(B(x))^2,x=0..15.4));Ratio:=CenterOfMass/A(0);