1. Minimize the perimeter of rectangles with area 25 cm^2 . Is there a maximum perimeter of rectangles with area 25 cm?
2. Find two numbers whose difference is 100 and whose product is a minimum.
3. If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
4. Find the point on the line 6x+y=9 that is closest to the point (-3,1)(Check your answer with the grapher)
5. Maximize the area of rectangles inscribed in the right triangle with lengths of sides 5, 12, and 13.
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Find the point on the line that is closest to the point (Check your answer with the grapher) ...
This solution looks and optimization and investigates how to calculate the closest point on a line. This solution is provided in two attached Word documents.
Business Calculus : Come up with an original optimization project.
I would like for you to complete an optimization project. Examples include: the optimum number of hours to study for an exam, the best time to leave for work to avoid traffic, the optimum speed to drive a dirt bike up a hill to achieve the longest jump.
I would like you to follow "The Process of Problem Solving" found below and complete each of the steps, A-E below. Be sure to follow each of the steps of the "The Process of Problem Solving" found below for this project. You can take a look at the example of "Meg" below, but I would like you to complete your own BRAND NEW OPTIMIZATION project.
Optimization problems involve finding the best way to do something in order to maximize or minimize some quantity. In business, the main goal is usually opitimization (especially maximizing profit or shareholders' wealth).
The Process of Problem Solving
A. Clarify the problem.
B. Formulate a model.
C. Solve/Analyze the model.
D. Validate the model and solution.
E. Draw conclusions and implement a solution.
This example is an analogy for how we propose to solve optimization and forecasting problems. In this analogy, the real world is like the ground, and the world of math is "up in the clouds." Stage A, clarifying the problem, happens on the ground. Stage B, formulating a model, is the connection (or translation) from the real world to the symbolic world, including gathering data. Stage C, solving the model, occurs in the world of math (up in the clouds, visually finding the most direct path through the maze and drawing it in). Stage D, validating the model, consists of flying lower again to check and make sure the model (the picture) accurately represents the real problem, and that the solution makes sense, making corrections and improvements where necessary. And stage E, drawing conclusions and implementing a solution, corresponds to coming back to the real world (the ground) and using the abstract solution (the map/picture with the solution path drawn on) to solve the real problem.
Here is a brief example of what I am looking for. Remember that it is only an example and I want you to complete something different than what is found below. I want you to COMPLETE YOUR OWN NEW OPTIMIZATION EXERCISE BY FOLLOWING STEPS A-E ABOVE:
Example: A student named Meg wanted to know what was the best amount of exercise for her to get. How could she figure this out?
Stage A: Clarify the Problem
The first step in solving this problem was to clarify exactly what the problem was. Was Meg trying to get into shape? Was she trying to lose weight? Was she trying to find the right balance between exercise and the rest of her life? Was she trying to find the level of exercise that would give her the most energy for her other activities? The initial description of this problem could have meant many different things, not only to different people, but even to Meg herself at different points in time.
Meg decided that she was primarily interested in the effect of exercise (running, in her case) on how she felt. But what did she really mean by "how she felt"? At first, she thought that perhaps this meant "happiness." Then she realized that, although her running did indeed affect her happiness, so did many other things that were much harder to control (relationships, school, etc.). The running might not have been a major factor in her happiness much of the time. She realized that what she was really interested in was how she felt physically, which she also thought of as her energy level. When she exercised too little, she felt lazy and lethargic, but if she exercised too much, she could feel exhausted, which would also correspond to a low energy level for other activites. She realized that her energy level might also be affected by her amount of sleep, but at the time, she was very careful to get eight hours of sleep every night, so that was not really a factor to be considered. Finally, she realized that she didn't always have time to run every day, so decided she would focus on determining the optimal amount of running each week.
Stage B: Formulate a Model
Now the question was how to gather data to help find the optimal amount of running. She decided to keep a log of her running, with the date, time of day when she started, and length of time (in minutes) running. But how could she record her energy level? She decided to use a subjective 1-100 scale, where 0 meant no energy (comatose) and 100 meant a virtually infinite energy level. She made up a scale as follows:
100: Infinite energy
90: Extremely high energy level
80: Very high energy level
70: High energy level
60: Pretty high energy level
50: Medium energy level
40: Moderately low energy level
30: Low energy level
20: Very low energy level
10: Extremely low energy level
0: No energy
The next question was: How often should she record her energy level? She could record it every day (say, just before going to bed at night), and average the values over a week. She could record it just before running every time, and again take some kind of average. Or she could record it at the end of each week, evaluating for the week as a whole. In some ways, the weekly assessment seemed closest to the idea of her problem, as long as she felt her memory was good enough to properly reflect the week overall. Meg felt that she could do this, so that was how she gathered her data. She could have decided to gather data in several ways, do different analyses, and compare the results, for even more confidence in her conclusions.
The following table gives the results of her data collection, with the number of minutes running and the corresponding energy level for each of ten weeks:
Running (min)380 465 420 240 275 390 730 355 440 605
68 90 80 30 40 70 45 60 75 85
The table is interesting and gives some clues about the optimal solution. Looking for the highest energy levels, they occur at 465, 420, and 605 minutes, so it would seem that the optimal level should be somewhere in that vicinity numerically. But it is hard to get a good feel for data just from a table. The old cliché says "a picture is worth a thousand words," and that is true in math as anywhere else. For data values, a good way to make a picture out of them is to draw a graph. In this case, we can draw a graph, with running time in a week (in minutes) on the horizontal axis and energy level (using the 0-100 scale) on the vertical axis. Each data observation (week) can then be represented as a data point on the graph. For instance, the fourth data point can be written as the ordered pair (240, 30) and plotted as the point that lines up with 240 on the horizontal axis and 30 on the vertical axis.
Stage C: Solve/Analyze the Model
The y value in the model is expressed in terms of the x value, and we could define it as y = the energy level (on the 1-100 scale) on average that results from running x minutes in a week.
Stage D: Validate the Model and Solution
Stage E: Draw Conclusions and Implement a Solution
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