Fall 2017 Midterm Exam PREDICT 400: Math for Modelers Points possible: 100 Description: The midterm exam will cover topics from sessions 1-4. Resources: The exam is completely open book. You may use course textbooks, materials provided on Canvas, graphing calculators (such as TI 83 or 84); but any more advanced calculators, Excel Solver, Web calculators, Web-graphic calculators, or simplex method calculators are not allowed. Programming languages other than Python are also not permitted. For questions that require calculations, all calculations should be shown, not just the final answer. This will allow for partial credit for those answers that might be set up correctly but have calculation errors. If variables are introduced, whether in a program or in a mathematical solution, you will need to explicitly state what the variables represent and clearly state the results based on the variable designations. For questions that specifically require Python, your code and output should be included to constitute work shown along with solutions. For questions that require graphs, only use Python. Python can be used for all questions unless instructed otherwise. For any problem the uses Python, your code and output must be provided. Restrictions: All answers are to be your work only. You are not to receive assistance from any other person. To complete the exam: 1. Answer all questions on the exam thoroughly. Create a single Microsoft Word document, including the question number, the question, your typed answer, and graphs if required. You may use Word’s equation editor to complete your answers. 2. Once you have completed your exam, save the file with a meaningful filename such as “Midterm” followed by your last name and return to the exam item where you downloaded the exam PDF, click View/Complete Assignment, and submit your document. 1. The cost to manufacture 1200 pairs of Nike’s bestselling Crossfit shoe, the Metcon X, is $66,000 and the cost to manufacture 2500 pairs is $111,500. Determine the linear cost function and the price Nike should sell the shoes to break even at 267 shoes (round to the nearest dollar). 2. In an effort to replace the destruction of trees, five regions of the US have been tracking the annual rate of trees being planted (per thousand) in each region and the annual rate of trees being destroyed (per thousand). Additionally, the tree population (in thousands) by region has been recorded for five different years. The data is given in the tables below. Use Python to determine the number of trees planted and the number destroyed in each of the years listed. Rate of Trees Being Planted Northeast Southeast Midwest Southwest West Year 1975 1985 1995 2005 2015 Northeast 365 471 622 803 1013 0.0341 0.0174 0.0185 0.0131 0.0096 Rate of Trees Being Destroyed 0.0115 0.0073 0.0056 0.0082 0.0105 Tree Population by Region Southeast Midwest Southwest 2036 285 226 2494 361 251 2976 441 278 3435 523 314 3827 592 344 West 460 485 499 514 522 3. Lucinda’s boutique sells three types of aromatherapy mixtures that each use essential oils, lavender, and chamomile. Scent A requires 3 oz. of essential oils, 2 oz. of lavender, and 1 oz. of chamomile. Scent B requires 3 oz. of essential oils, 1 oz. of lavender, and 2 oz. of chamomile. Scent C requires 2 oz. of essential oils, 1 oz. of lavender, and 2 oz. of chamomile. Determine how many of each scent can be made with 915 oz. of essential oils, 465 oz. of lavender, and 525 oz. of chamomile. 4. Ed’s lawn supply store has 165kg of rye grass seed and 93kg of bluegrass seed. He plans to sell two different grass seed mixes. One mix will contain half rye grass seed and half bluegrass seed and will sell for $8.25 per kg. The other mix will contain ¾ rye grass seed and ¼ bluegrass seed and will sell for $7.50 per kg. Use Python to determine how many kilograms of each seed mixture he should prepare for the maximum revenue, as well as that revenue. 5. Camp Christopher wants to hire counselors and assistants to fill its seasonal staffing needs at a minimum cost. The average monthly salary of a counselor is $2400 and the average monthly salary of an assistant is $1100. The camp can hire up to 35 staff members and needs at least 20 to run properly. They must have at least 10 assistants and may have up to 3 assistants for every 2 counselors. Using Python, graph the feasible region and determine the number of counselors and assistants the campaign should hire. 6. Louise has decided to plan a monthly exercise program to include yoga, Crossfit, and swimming. She would like to limit her exercise routine to at most 28 hours per month. Her plan would be to spend at most 9 hours swimming, and limit Crossfit to no more than the total hours spent swimming and doing yoga. She can burn 200 calories per hour of yoga, 597 calories per hour of Crossfit, and 263 calories per hour of swimming. How many hours should be spent for each activity to maximize the number of calories burned and how many calories will she burn? 7. Goodyear Commercial Retread facility recycles worn or damaged tires to retread two sizes of tires, medium and large, for tractor trailers. To fill one client’s order, they will use a combination of three machines to manufacture the retreads and will need to produce at least 54 medium tires and 59 large tires. In one hour, Machine I can produce 2.5 medium tires and 5 large tires, Machine II can produce 4.5 medium tires and 3 large tires, and Machine III can produce 5 medium tires and 10 large tires. The cost per hour to operate Machine I, II, and III are $8, $9, and $10, respectively. There are two combinations of using the three machines that produce the same minimum cost. What are those two combinations and what is the minimum cost? 8. A company is creating a new fertilizer additive for lawn seed, which will be a mixture of nitrogen, phosphate, and potash. Based on their research, the total amount fertilizer added must be at least 14 oz. per 5 lb. bag of lawn seed, but should not exceed 20 oz. per 5 lb. bag. At least ¼ oz. of nitrogen must be used for every ounce of phosphate, and at least 1 oz. of potash must be used for every ounce of nitrogen. The costs per ounce of nitrogen, phosphate, and potash are $0.30, $0.18, and $0.54, respectively. Use Python to determine the mixture of the three ingredients in a 5 lb. bag that minimizes costs, as well as that cost. 9. A survey of 4500 people was conducted to collect data on family history of high blood pressure (HBP), high cholesterol (HC), and heart disease (HD) among Caucasian women. The responses showed that 1621 had a family history of all three, 1829 had a family history of HBP and HC, 695 had a family history of HBP only, 2710 had a family history of HBP, 1779 had a family history of HD and HC, 2619 had a family history of HC, and 2537 had a family history of HD. How many of the women surveyed had no family history of any of these? 10. A new type of test to determine the presence of lead in drinking water is successful 97.5% of the time when lead is present. However, the same test indicates lead is present 0.3% of the time when it is not present. One percent of the homes in Cleveland actually have lead present in the drinking water. Determine the probability that a home in Cleveland has lead in the drinking water given that the test indicates the presence of lead.
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