You will write a report in memo format (sample provided) which fully reflects your work. The mathematical content is of course most important – with all work shown. Analysis of the mathematical content is then to be completed that explains the work to a perhaps lay audience and draws two or more conclusions/recommendations. The questions posed in the project should be answered/addressed in the order posed. Grammar and spelling are also important. The submission will need to be typed and in a format that has a mathematics editor for your mathematical work. Any needed graphs should also be included.
Project 1
In this project, you play the role of a consultant to a plastics manufacturer. In particular, you will use a
number of tools from Calculus III (including solids of revolution, and volume) to advise a client on the construction of plastic cups whose form is roughly that of paraboloid of revolution.
Suppose that we construct objects in the shape of a solid of revolution so that a cross-section of the solid has a parabola as a bounding curve. Such a solid is called a paraboloid of revolution. We will construct the portion of the solid z=1/4p(x^2+y^2) bounded by the plane z=k, for a constant k>0. The circular section cut by the plane is called the aperture and the distance between the origin and the plane is called the depth.
Find the equation of the paraboloid of revolution whose aperture diameter at depth 𝑐𝑐 cm is 10 cm.
For the solid in Question 1, find the coordinates of the focus of the paraboloid.
Find the volume of this solid.
Construct a formula for the cost of this solid if plastic is 0.02 cents per cubic centimeter.
Construct a formula for the cost of a cup if it has aperture diameter at depth 𝑐𝑐 cm is 10 cm, and the wall of the cup must be between 0.5 and 1 cm.
Using Question 5, find the cost (to the nearest cent) per cup for cup wall thickness 0.6 cm and depth 7.5 cm.