A86 APPENDIX Review 124. General Chemistry For a certain ideal gas, the volume V (in cubic centimeters) equals 20 times the temperature T (in degrees Celsius). If the temperature varies from 80° to 120 C, ° inclusive, what is the corresponding range of the volume of the gas? 125. Real Estate A real estate agent agrees to sell an apartment complex according to the following commission schedule: $45,000 plus 25% of the selling price in excess of $900,000. Assuming that the complex will sell at some price between $900,000 and $1,100,000, inclusive, over what range does the agent’s commission vary? How does the commission vary as a percent of selling price? 126. Sales Commission A used car salesperson is paid a commission of $25 plus 40% of the selling price in excess of owner’s cost. The owner claims that used cars typically sell for at least owner’s cost plus $200 and at most owner’s cost plus $3000. For each sale made, over what range can the salesperson expect the commission to vary? 127. Federal Tax Withholding The percentage method of withholding for federal income tax (2022) states that a single person whose weekly wages, after subtracting withholding allowances, are over $312.50, but not over $961.06, shall have $1100 plus 12% of the excess over $312.50 withheld. Over what range does the amount withheld vary if the weekly wages vary from $500 to $700. Source: Publication 15-T (2023), Federal Income Tax Withholding Methods 128. Aerobic Exercise For most healthy adults, the U.S. Department of Health and Human Services recommends at least 150 minutes of moderate aerobic activity a week. The guidelines suggest that you spread out this exercise during the course of a week. For the first 5 days of the week, you exercise 25, 35, 0, 40, and 15 minutes. How long should you exercise over the last two days of the week to meet the minimum number of recommended minutes? Source: Mayo Clinic 129. Cellular Phone Rates T-Mobile offers an Essentials rate plan for one phone line with autopay that costs $60 per month. The Essentials plan includes unlimited talk and text within the United States with international calls costing $0.25 per minute. One customer owns a business with clients in Europe. If their monthly bills, over a one-year period, ranged from a low of $69.50 to a high of $140.75, over what range (in minutes) did international calls vary? Source: T-Mobile, April, 2023 130. Uber Rates Uber customers have coined the term “surge pricing” to describe the fact that a person can use an Uber to travel the same route and pay a higher price based on the time of day the route is traveled. One customer performed an experiment and used an Uber to travel 23 miles from their home to the airport. Five separate trips, at varying times and days, had costs of $20.93, $28.23, $40.44, $22.67, and $34.55. What is the range of the cost-per-mile for the five trips? Source: Uber 131. Fat Content Suppose that you order a small McCafe™ chocolate shake (14 g of fat) and a Quarter Pounder With Cheese™ (26 g of fat) at McDonald’s. How many chocolate chip cookies can you eat (8 g of fat) and still keep the total fat content of your meal to no more than 64 g? 132. Counting Calories Suppose that you order a Bacon, Sausage, Egg, & Cheese Biscuit (634 calories) and a Fat Free Milk (96 calories) at Burger King. How many orders of MOTTS Apple Sauce can you eat (50 calories) and still keep the total calories of your meal to no more than 830 calories? 133. Computing Grades In your Economics 101 class, you have scores of 68, 82, 87, and 89 on the first four of five tests. To get a grade of B, the average of the first five test scores must be greater than or equal to 80 and less than 90. (a) Solve an inequality to find the least score you can get on the last test and still earn a B. (b) What score do you need if the fifth test counts double? What do I need to get a B? 68 87 82 89 134. IQ Tests A standard intelligence test has an average score of 100. According to statistical theory, of the people who take the test, the 2.5% with the highest scores will have scores of more than 1.96σ above the average, where σ (sigma, a number called the standard deviation) depends on the nature of the test. If 12 σ = for this test and there is (in principle) no upper limit to the score possible on the test, write the interval of possible test scores of the people in the top 2.5%. 135. Arithmetic Mean If a b, < show that a a b b 2 . < + < The number a b 2 + is called the arithmetic mean of a and b. 136. Refer to Problem 135. Show that the arithmetic mean of a and b is equidistant from a and b. 137. Geometric Mean If a b 0 , < < show that a ab b. < < The number ab is called the geometric mean of a and b. 138. Refer to Problems 135 and 137. Show that the geometric mean of a and b is less than the arithmetic mean of a and b. Explaining Concepts 139. Make up an inequality that has no solution. Make up an inequality that has exactly one solution. 140. The inequality x 1 5 2 + <− has no real solution. Explain why. 141. Do you prefer to use inequality notation or interval notation to express the solution to an inequality? Give your reasons. Are there particular circumstances when you prefer one to the other? Cite examples. ‘Are You Prepared?’ Answers 1. -4 -2 0 2. False
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