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tags: mth201, templates
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# AEP 6: Applied Optimization

## What this AEP is about

This AEP consists of three applied optimization problems in the style of what we've seen in class.

## Prerequisites and tech requirements

You will be ready to begin this AEP following **Module 9.**

**Technology used in this AEP:** Desmos, for doing a graphical check of your models and solutions; you'll also be able to use Wolfram|Alpha to automate some of the algebra and other pre-calculus math.

## Background and setup

There's no background for this AEP other than what you learned in class. However, you'll need to take care to submit **a very clear, clean, and professional looking writeup that includes a complete solution that is clearly commnunicated**. To help you with this, here is a solution to one of the exercises from Section 3.4 from the _Active Calculus_ book. This solution is in a written form that, if it were submitted as an AEP, would get an "E" mark. **Use this as your guide.**

https://docs.google.com/document/d/1vG8m2x3uJg_zBZggL4QNoOj3iabK7ocS9P_C17SGYgY/edit

In particular, please see the grading criteria for this AEP below, as there are a few items that **must** appear in your solutions in order to make these meet the standards.

Comments are turned on with this document, so if you have a question or a comment, just highlight the text, then click *Insert* and then *Comment*.

## Special Offer!

**Earning an "E" on this AEP will level you up on Learning Target 12**. That is, an "E" on this AEP will count the same as doing acceptable work on a Checkpoint problem for Learning Target 12. So you can earn Mastery on Learning Target 12, for example, by doing acceptable work on a Checkpoint problem and then earning an E on this assignment, or vice versa.

## AEP Tasks

**Choose exactly one of the following problems and give a complete, correct, and clearly-communicated solution.**

1. Two vertical poles of heights 50 ft and 100 ft stand on level ground, with their bases 200 ft apart. A cable that is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. What is the minimum possible length of cable required?
2. A rectangular box with a square bottom and closed top is to be made from two materials. The material for the side costs $2.50 per square foot and the material for the top and bottom costs $5.00 per square foot. If you are willing to spend $25 on the box, what is the largest volume it can contain?
3. A company is designing propane tanks that are cylindrical with hemispherical ends. Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the ends are more expensive to make, costing $4 per square foot, while the cylindrical barrel between the ends costs $1.75 per square foot. Use calculus to determine the minimum cost to construct such a tank.

Again, **do only one of these**. If you submit substantial work on more than one problem, the submission will be marked "X" and returned without comment.

## Grading criteria and submission instructions

Please refer to the [overall quality standards for AEPs](https://hackmd.io/@rtalbert235/HkSbMs2Av) at the link (and posted to the *AEPs* area on Blackboard) first, and make sure your work meets all these criteria. In addition to the overall standards, this AEP has some **very specific standards** that must be met. Skipping any of these will result in a grade of "P" or "X":

- **Your solutions must not just be computations**. There must be significant English narrative, consisting of clear and correctly constructed sentences, present to guide the reader through the solution. Again, [the example solution above](https://docs.google.com/document/d/1vG8m2x3uJg_zBZggL4QNoOj3iabK7ocS9P_C17SGYgY/edit) shows what this looks like.
- **Each solution must clearly state in some way what the basic variables are, as well as what they represent and what their units are; solutions must also clearly state the target variable (the quantity you're trying to optimize) and what its units are.** Clearly state these variable *in English* and not just as labels on a diagram.
- **Each solution must produce a formula in one variable for the target quantity and explain all the steps used to arrive at that formula.** This includes identifying any constraints in the problem and how you use the constraint to simplify the target formula to one variable.
- **Your solution must state the domain of the target function and explain in detail how you arrived at that domain.** Especially, if your domain is a closed interval, explain how you got it.
- **Each solution must use Calculus techniques to find the optimum value begin asked for.** You may not simply estimate the value from a graph or use any non-calculus alternatives, unless it's to check work that you did using calculus.
- **Each solution must include mathematical reasoning that proves that the answer you provide is truly the absolute maximum or minimum of the function you are optimizing.** You must use one of the three techniques discussed and practiced in class --- the Extreme Value Theorem, the modified First Derivative Test, or the modified Second Derivative Test --- and explain why you chose your method. You may not simply point to a graph.

Finally: **You are allowed to use computer tools to find derivatives and do algebraic simplification on this AEP.** For example you may use [Wolfram|Alpha](http://wolframalpha.com) to find derivatives and solve equations. **If you do, you must include a link to your calculations.** You do this just by copying and pasting the URL. For example, here's a Wolfram|Alpha link to an algebra computation: https://www.wolframalpha.com/input/?i=expand+%28x%2Bh%29%5E5

Since you are allowed to use computer tools to do computations, note again that **the emphasis of evaluating your work is firmly on the quality of your solution and how well you communicate it**. It needs to be complete, neatly formatted, and readable, like a good solution in a textbook. Use the sample solution mentioned above as your guide.

Please [see the syllabus](https://hackmd.io/@rtalbert235/SJ5fDZIAv#How-are-individual-assignments-graded) for how grades of **P** (Progressing) and **X** (Not Assessable) are assigned.

To submit this AEP:

- Create a **typed document** using a word processor utilizing an equation editor for any significant mathematical notation. **Submissions that are handwritten or contain any handwritten work without prior permission will be graded "X" and returned without comment.**
- Save the document as a PDF. Please **do not submit Word (.doc, .docx) or Open Office (.odt) files** as these do not always render mathematical notation correctly on screen.
- Upload the PDF to the appropriate AEP assignment in the *AEPs* area on Blackboard. **Remember to hit the "Submit" button after uploading**, otherwise your work is merely on the Blackboard server and not in the instructor's grading queue.

Once graded, the grade (E, M, P, X) will appear in your gradebook. You can then open the submission to view feedback left by the professor. These appear as text comments on the page or as general comments next to the grade. Grades of E or M may not have much feedback. Grades of P or X *always* have feedback, so please look carefully for this.

For information on revisions of AEPs, please [consult the syllabus](https://hackmd.io/@rtalbert235/SJ5fDZIAv#How-do-I-revise-and-resubmit-my-work).

[](https://hackmd.io/0hGsJdDIQ32XTMVuy2I1tw)