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Homework section 2.2 In problems 1-10, translate each sentence into a mathematical equation. Be sure to define the meaning of all symbols. 1) The area of a circle is the product of the number radius. and the square of the 2) The circumference of a circle is the product of the number radius. and twice the 3) The area of a square is the square of the length of a side. 4) The perimeter of a square is four times the length of a side. 5) Force equals the product of mass and acceleration. 6) Pressure is force per unit area. 7) Work equals force times distance. 8) Kinetic energy is one-half the product of the mass and the square of the velocity. 9) The total variable cost of manufacturing x dishwashers is $150 per dishwasher times the number of dishwashers manufactured. 10)The total revenue derived from selling x dishwashers is $250 per dishwasher times the number of dishwashers sold. 11)A total of $10,000 is to be divided between Sean and George with George to receive $3000 less than Sean. How much will each receive? 12)Going into the final exam, which will count as two- thirds of the final grade, Mike has test scores of 86, 80, 84, and 90. What score does Mike need on the final exam in order to earn a B, which requires a score of 80? What does he need to earn an A, which requires an average of 90? 13)The suggested list price of a new car is $12,000. The dealer’s cost is 85% of the list. How much will you pay if the dealer is willing to accept $100 over the cost for the car? 14)A wool suit, discounted by 30% for a clearance sale, has a price tag of $399. What was the suits original price? 15)Wendy, a loan officer at a bank, has $1,000,000 to lend and is required to obtain an average return of 18% per year. If she can lend at the rate of 19% or at the rate of 16%, how much can she lend at the 16% rate and still meet her requirement? Principal Rate % Time Interest Lend at 18% interest Lend at 16% interest Total 16)A motorboat heads upstream on a river that has a current of 3 miles per hour. The trip upstream takes 5 hours, and the return trip takes 2.5 hours. What is the speed of the motorboat? (Assume that the motorboat maintains a constant speed relative to the water. V mph Upstream Downstream t hrs s 17)A Metra commuter train leaves union station in Chicago at 12 noon. Two hours later, an Amtrak train leaves on the same track, traveling at an average speed that is 50 miles per hour faster than the Metra train. At 3 pm, the Amtrak train is 10 miles behind the commuter train. How fast is each going? V mph t hrs s Metra Amtrack 18)Two cars enter the Florida Turnpike at Commercial Boulevard at 8:00 AM, each heading for Wildwood. One car’s average speed is 10 miles per hour more than the others. The faster car arrives at wildwood at 11:00AM, ½ hour before the other car. What is the average speed of each car? How far did each travel? V mph Slower Car Faster Car t hrs s 19)How much water should be added to 1 gallon of pure anti freeze to obtain a solution that is 60% antifreeze? gallon solution % antifreeze gallon antifreeze Antifreeze Water Mixture 20)The cooling system of a certain foreign made car has a capacity of 15 liters. If the system is filled with a mixture that is 40 % antifreeze, how much of the mixture should be drained and replaced by pure antifreeze so that the system is filled with a solution that is 60% antifreeze? liter solution 40% antifreeze sol. Pure antifreeze Mixture % antifreeze liter antifreeze 21)How much water must be evaporated from 32 ounces of a 4% salt solution to make a 6% salt solution? Oz. solution % Salt Oz Salt 4% salt solution Water Mixture 22)How much water must be evaporated from 240 gallons of a 3% salt solution to produce a 5% salt solution? Oz. solution 3% salt solution Water Mixture % Salt Oz Salt 23)The monthly payment p on a mortgage varied directly with the amount borrowed B. if the monthly payment on a 15-year mortgage is $8.99 for every $1000 borrowed, find a linear function that relates the monthly payment p to the amount borrowed B for a mortgage with the same terms. Then, find the monthly payment p when the amount borrowed B is $175,000. In problems 24 – 29, examine the scatter diagram and determine whether the type of relation, if any, that may exist is linear or nonlinear. 24) 25) 26) 27) 28) 29) In problems 30, 31, a) Draw a scatter diagram. b) Select two points from the scatter diagram and find the equation of the line containing the points selected. c) Graph the line found in part (b) on the scatter diagram. d) Use a graphing utility to find the line of best fit. e) Use a graphing utility to graph the line of best fit on the scatter diagram. 30) x 3 4 5 6 7 8 9 y 4 6 7 10 12 14 16 31) x -30 -27 -25 -20 -14 y 10 12 13 13 18 32) An economist wishes to estimate a linear function that relates per capita consumption expenditures C and disposable income I. Both C and I are measured in dollars. The following lowing data represent the per capita disposable income (income after taxes) and per capita consumption in the United States for 1990 to 1998. Year Per Capita Disposable income (I) Per capita consumption (C) 1990 15,695 14,547 1991 16,700 15,400 1992 17,346 16,035 1993 18,153 16,951 1994 19,711 18,419 1995 20,316 19,061 1996 21,127 19,938 1997 21,871 20,807 1998 22,212 21,385 Let I represent the independent variable and C the dependent variable. a) Using a graphing utility to draw a scatter diagram. b) Use a graphing utility to find the line of best fit to the data. Express the solution using function notation. c) Interpret the slope. The slope of this line is called the margin propensity to consume. d) Predict the per capita consumption in a year where disposable income is $21,500. e) What is the income in a year where consumption is $17,200? 33) The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income. The following data represent the annual income, I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 35 years. Annual Income (I) Loan Amount (L) 15,000 44,600 20,000 59,500 25,000 74,500 30,000 89,400 35,000 104,300 40,000 119,200 45,000 134,100 50,000 149,000 55,000 163,900 60,000 178,800 65,000 193,700 70,000 208,600 Let I represent the independent variable and L the dependent variable. a) Use a graphing utility to draw a scatter diagram. b) Use a graphing utility to find the line of best fit to the data. c) Graph the line of best fit on the scatter diagram drawn in part (a). d) Interpret the slope of the line of best fit. e) Determine the loan amount that an individual would qualify for if her income was $42,000.