Optimization Project

$20$ points

Everyone should work on this project in groups of $2$ or $3.$ This is a requirement.

Each group should submit one typed paper copy. It should be turned in to me before or after one of our class periods, or during my student$/$office times no later than $11$:$30$ AM on Wednesday, December $20.$ This copy should have all group members' full names typed on the top of the first page. There is a risk in simply sliding the assignment under my door, so make sure to hand the copy to me$.$ Each member of the group is responsible for ensuring that one copy is turned in for your group. To encourage students to start their work early, projects turned in by Monday, December $18$ at $11$:$30$ AM will receive two bonus points on their project $($These should be siven to me and not simply put under my door.$).$ Each person should also submit a pdf copy of the project on Canvas by $11$:$30$ AM Wednesday, December $20.$ I will grade the paper copy, but also want the pdf submitted as a record.

The unit price $($in dollars$)$ for making each of $x$ bikes each week $)x(25$ is $p(x)=1,200-0\mathrm{.}4x^2.$

The cost $($in dollars$)$ for making $x$ bikes weekly is given by $C(x)=3,000+840x-0\mathrm{.}1x^3.$

Find the price that will maximize profit each week.

Will the price found in part $1$ also maximize the revenue for the week? Is yes, show why. If this is not the price that will maximize the weekly revenue, find that price.

For a business, is it more desirable to maximize the profit or the revenue? Explain why.

The Ground Rules

This is a group project. You should not ask people outside your group for help with these problems. Except for exceptional circumstances, each group member will receive the same grade. Each student should try to solve as much of this as possible prior to meeting with your group so that they can contribute to the group. If all members have problems, you can ask me for a hint.

Write$-$up

Each group will create a typed report. Use $12-$point font and $1\mathrm{.}5$ spacing between lines. The report must include the names of all team members and should be $1$ or $2$ pages in length.

Write the text of your paper in complete sentences. Begin with introducing the problem. Use the numbering given to organize the work, You do not need to write the mathematics in sentences, but first describe what you will do in each major step with text. You can hand write in sign charts and any math notation you cannot easily type but be sure to do this neatly in pen.

The report should be prepared so that it would be understood by students with backgrounds similar to those of the students in the class. No prior knowledge of the project should be necessary to fully understand what the assignment was. The report should not contain a reproduction of this sheet; the report must be understandable simply because it is clearly written. Each member of the team should work on the write$-$up and proofing the work prior to submitting. Grades will not be posted until all groups have submitted their projects and all have been graded. You will be graded on both the mathematics and the write$-$up$.$