# Multivariable Calculus Problem Set

1) Represent the volume (don’t have to evaluate) between the surfaces of an inverted circular cone with axis along z-axis and a sphere of radius a, centered at the point (0,0,a), using a triple integral in,
(a) Spherical coordinates.
(b) Cylindrical coordinates
(c) Cartesian coordinates
Assume that the cone has radius of cross-section (parallel to x-y plane) equal to the height of the cross-section from the x-y plane. Start each case drawing a figure (No points without figure) to substantiate the limits of your integral.

2)Consider the surface S: z^2 = a(x^2+y^2), z ≥ 0 for some constant a > 0. Find the dimensions of this shape S such that the volume contained in it is 36π cm^3 and it has a minimum surface area.
(a) Identify the shape of the surface. What are the relevant dimensions that can completely define the surface? (e.g. length, width, and height if its a cuboid; radius and height if its a cylinder). Draw a figure.
(b) What is the function being minimized in terms of the dimensions in the previous question?
(c) What is the volume contained in the shape in terms of the dimensions?
(d) Set up the equations that allow you to find the dimensions that minimize the surface area.
(e) Reduce the system of equations to represent it in terms of one variable and solve to find the dimensions of the cone.

• I am trying to answer your question, but consider extending the deadline. These questions are difficult and need more time.

• I am sorry, that is when it is due. Anything helps, really just need to see the work behind the answer