- #1
Michels10
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Homework Statement
A toy manufacturer wants to create a toy ice cream cone by fitting a sphere of radius 4 cm inside a cone with a height of 8 cm and radius of the base of 3 cm. The base of the cone is concave, but the rest of the cone is solid plastic so that with the sphere attached there is no hollow space inside. The sphere and cone are constructed from special materials so that the density of the sphere is proportional to the distance from the tip of the cone, with constant of proportionality 1.4, and the density of the cone is proportional to the distance from its vertical axis, with constant of proportionality 1.8. Find the mass and center of mass of the ice cream cone. (Hint: decide on a position for the whole ice cream cone. If the center of the sphere is the origin, then the tip of the cone cannot also be at the origin. It is a good idea to have the cone oriented vertically the way you would normally hold one.) Also, provide a plot showing the entire ice cream cone.
Note from professor: Instead of dealing with these
integrals, I will allow you to change the density functions given so
that instead the densities for both the cone and sphere are
proportional to the square of the distance described. With these
simpler density functions (without the square roots), the integrals
should be much simpler to solve.
Homework Equations
Sphere: z = x^2 + y^2 + z^2
Cone: z = sqrt(x^2 + y^2)
Vsphere = 4/3*pi*r^3
Vcone = (pi*h*r^2)/3
density = mass/volume
center of mass:
x(bar) = Myz/m
y(bar) = Mxz/m
z(bar) = Mxy/m
where M is the moment about a coordinate plane.
The Attempt at a Solution
Not exactly sure where to start... If I have no mass how can I begin to solve for this? Should I start by solving for the volume that the sphere+cone take up?
Any help is greatly appreciated!
Thanks,
Michels10