MATH , Exam 1 Review Fall 2009

Transcription

1 MATH , Exam Review Fall 2009 Solutions I. Venn Diagrams and Arguments For problems -5, draw and label a Venn diagram for the given statement.. Some philosophers did not live in Greece. Solution: A = Philosophers who didn t live in Greece B = People who did live in Greece but were not philosophers C = Philosophers who did live in Greece D = Everyone else 2. Herman hates spiders. Solution: A = Spiders B = Things that Herman hates C = Things that Herman doesn t hate We could also illustrate this statement using a Venn diagram of two disjoint sets. What could you choose for these two sets?

2 3. Cheaters never prosper. Solution: A = Cheaters B = People who prosper C = People who don t prosper but aren t cheaters We could also illustrate this statement using a Venn diagram of a subset of another set. What could you choose for these two sets? 4. Some teenagers are responsible drivers. Solution: A = Teenagers who drive irresponsibly B = Responsible drivers who aren t teens C = Teenagers who drive responsibly D = Everyone else 5. All widows are women. Solution: A = Widows B = Women C = People who aren t women For problems 6-0, decide whether an argument is inductive or deductive. If it is inductive, explain why it is strong or why it s not. If the argument is deductive, draw the appropriate Venn diagram and decide whether it is valid or sound.

3 6. Premise: Anne is not in any pain. Premise: Her blood pressure is normal. Conclusion: She is in good health. Solution: This is an inductive argument because the premises only prove that Anne might be healthy. Since there are a significant number of diseases/health conditions that do not immediately cause pain or negatively affect blood pressure, this argument is relatively weak. 7. Premise: If you swallow food and talk at the same time, you will choke. Premise: Ted is choking. Conclusion: Ted must have swallowed food and talked at the same time. Solution: This problem is essentially the same as example 6 on page 60 of your textbook. The argument is a deductive argument that is not valid and not sound. 8. Premise: All great basketball players must wear expensive shoes. Premise: Shaquille O Neal is a great basketball player. Conclusion: He must wear expensive shoes. Solution: This problem is essentially the same as example 5 on page 60 of your textbook. The argument is a deductive argument that is valid but not sound. 9. Premise: Every winter it has snowed in Utah for all recorded history. Conclusion: It will snow in Utah this coming winter. Solution: This is an inductive argument because the premises only prove that the conclusion is likely to be true. Since the premises support the conclusion fairly convincingly, I would say that this argument is strong. 0. Premise: If you like the book, then you ll love the movie. Premise: You did not like the book. Conclusion: You will not love the movie. Solution: See example 7 on page 6 of your textbook. The argument is a deductive argument that is not valid and not sound. II. Unit Conversions. Convert 2 decimeters into decameters. Solution: Remember that decimeter = 0 meters and decameter = 0 meters. 0 2 meter decimeters decimeters decameters meters = decameters =.2 decameters 0

4 So 2 decimeters is.2 decameters. 2. How many joules are in one kilowatt-hour ( watt = joule/second)? Solution: We first notice that joule is equal to watt-second; just multiply top and bottom of the hint by second. Now, convert kilowatt-hours into kilowatt-seconds kilowatt- hour 60 minutes seconds hour 60 = 60(60) kilowatt-seconds = 3600 kilowatt-seconds minutes Next, convert kilowatt-seconds into watt-seconds 3600 kilowatt-seconds watts 000 kilowatt = 3600(000) watt-seconds = watt-seconds Since watt-seconds is equivalent to joules, there are joules in kilowatt-hour. 3. Convert 25 miles per hour into meters per second ( mile =.6 kilometers). Solution: First, convert miles per hour into meters per hour (remember that meter =,000 kilometers). 25 miles hour.6 kilometer miles 000 meters kilometer = 25(.6)000meters hour Second, convert meters per hour into meters per second. 40, 000 meters hour hour 60 minutes minute 60 seconds So 25 miles per hour is about meters per second. 4. Convert 8 grams per milliliter into grams per 00 milliters. Solution: We have grams 8 millimeter = 8 grams millimeter = 40, 000 meters hour 40, 000 meters = 60(60) second.meters second = 800 grams 00 milligrams In other words, 8 grams per millimeter is 800 grams per 00 millimeters. 5. Suppose you bought an imported Italian car but are concerned about its gas mileage. When you fill up its 5.8 gallon gas tank, the odometer reads 500 kilometers, and when you need to refuel, the odometer reads 2,0.26 kilometers. What is your car s

5 gas mileage, in miles per gallon? Remember that there are miles in one kilometer. Solution: Your car can drive 2, =60.26 kilometers before refueling. Converting this to miles gives kilometers miles kilometer = miles So your car can travel miles per 5.8 gallons of gas. That is, miles 5.8 gallons = miles gallons. This means that your car gets about 24 miles per gallon. 6. Scientists come through for once and discover that it takes 47 licks to get to the center of a Tootsie roll pop. If you lick at the rate of 53 licks per minute and your friend licks at the rate of 79 licks per minute, how many more seconds will it take you to finish your lollipop than your friend? Solution: First, let s calculate how many minutes it takes you to get to the center of a Tootsie roll pop. 47 licks minute 53 licks = minutes = minutes Second, let s calculate how many minutes it takes your friend to get to the center of a Tootsie roll pop. 47 licks minute 79 licks = minutes = minutes This means that it takes you = more minutes than your friend. In seconds, this is minutes 60 seconds minutes = (60) seconds = seconds So it takes you more seconds to get to the center of a Tootsie roll pop. 7. A new coal-burning power plant can generate.5 gigawatts (billion watts) of power. Burning kilogram of coal yields about 450 kilowatt-hours of energy.

6 (a) How much energy, in kilowatt-hours, can the plant generate each month? (Assume that there are 30 days in a month.) Solution: In kilowatts,.5 gigawatts (i.e.,.5 billion= watts) is watts In hours, a 30-day month is kilowatt 0 3 watts =.5(09 ) kilowatts = kilowatts days 24 hours day = 30(24) hours = 720 hours So the energy the plant generate each month is kilowatts 720 hours = kilowatt-hours (that is, 080 million kilowatt-hours). (b) How much coal, in kilograms, is needed by this power plant each month to generate this much energy? Solution: Based on our solution to part (a), we can can rephrase the question as follows: How much coal in kilograms needs to be burned to give us kilowatt-hours? kilowatt-hours kilograms of coal = kilograms of coal 450 kilowatt-hours 450 = kilograms of coal So 2.4 million kilograms of coal are needed. (c) If a typical home uses 000 kilowatt-hours per month, how many homes can the power plant supply? Solution: We have kilowatt-hours home = homes 000 kilowatt-hours 000 = homes So the power plant can supply,080,000 homes each month. 8. China has a population of,32,85,888 people (July 2007 estimate). If the birth rate is 3.45 births per,000 population per year and the death rate is 7 deaths per,000 population per year, what will China s population be next year?

7 Solution: First, let s calculate how many people will die this year., 32, 85, 888 people 7 deaths, 32, 85, 888(7) 000 people = 000 Second, let s calculate how many people will be born this year., 32, 85, 888 people births people =, 32, 85, 888(3.45) 000 deaths 9, 252, 963 deaths births 7, 778, 908 births So China will have about, 32, 85, 888 9, 252, , 778, 908 =, 330, 377, 833 people next year. 9. Suppose you own an aquarium 36 in. long, 5 in. wide, and 8 in. high. How many gallons of water can it hold? (Hint: There are 7.48 gallons in one cubic foot.) Solution: The volume of the aquarium is 36 in.(5 in.)(8 in.) = 9, 720 in. in. in. = 9, 720 in. 3. In gallons, this is 9, 720 in. 3 ft in gallons = ft 3 So the aquarium can hold about 42 gallons of water. 9, 720(7.48) 2 3 gallons 42 gallons 0. You re a paramedic behind the wheel of an ambulance and need to drive to the scene of an accident 34 blocks away. If you need to get there within 5 minutes, how fast do you need to drive, in miles per hour? (Assume that there are 2 blocks in one mile.) Solution: First, convert 34 blocks into miles. 34 blocks mile 2 blocks = 34 2 Second, convert 5 minutes into hours. 5 minutes hour 60 minutes = 5 60 miles = miles hours =.25 hours Finally, divide the number of blocks in miles by the number of minutes in hours miles.25 hours = miles hour So the we need to drive faster than about miles per hour. manageable.) (This seems fairly

8 III. Percentages Suppose a typical McDonald s meal consists of a Big Mac burger, medium fries, and a medium chocolate shake. Use the table provided below for problems??-??. Food Calories Fat (in grams) Big Mac Fries Shake A typical McDonald s meal contributes what percentage of my recommended daily Calorie intake. Assume that my recommended daily Calorie intake is 2300 Calories. (This figure is not unusual.) Solution: The total amount of Calories in a McDonald s meal is =, 78 Cal. So a McDonald s meal contributes of my recommended Calorie intake., 78 00% = % 75% 2, Suppose I order a typical McDonald s meal except that I ask for the Big Mac sauce to be left off my Big Mac. In this case, my Big Mac (without its signature sauce) contributes about 36% of the fat in my meal. So how much fat does a Big Mac with the sauce left off have? Solution: Let x be the amount of fat in a Big Mac (without the sauce). Then we know that x = 36% of the fat in my meal =.36(the amount of fat in my fries, shake, and Big Mac) =.36( x) =.36(4) +.36x Subtracting.36x from both sides gives x.36x =.36(4).64x =.36(4) Divide both sides by.64: x =.36(4) So a Big Mac without the sauce contains about 23 grams of fat.

9 3. If a Big Mac contains 28.7% less Calories than a Whopper (with cheese) from Burger King, how many Calories does a Whopper contain? Solution: Let x be the Calories in a Whopper. Then we have that Calories in a Big Mac = 563 = x 28.7 % of x = x.287x =.73x Dividing by.73 gives us that x = 563/.73 = Cal are in a Whopper (with cheese). 4. Last year, oil paintings comprised 73% of a certain art gallery s collection. This year, oil paintings comprise 70% of the gallery s collection, but the total number of the gallery s art pieces is 3% more than the number last year. By what percent has the number of oil paintings changed since last year? Solution: Let T be the total number of pieces in the art gallery last year, so the total number of oil paintings last year was.73t. Since the total number of the gallery s art pieces is 3% more than the number last year, the total number of art pieces this year is.03t. So the number of oil paintings this year is 70% of the art pieces this year =.70(.03(T )) This means that the percent by which the number of oil paintings has changed since last year is new number of oil paintings - old number of oil paintings old number of oil paintings That is,.70(.03(t )).73T 00%.73T We can cancel out the T s and simplify to get our answer:.70(.03) %.233% 00% 5. If a 34-ounce can of olives is 83% more money than a 2-once can of olives, which is a better deal? Explain. Solution: The larger can of olives has 34 2 = 22 ounces more than the smaller can of olives, so the 34-ounce can contains 22 00% 83.3% more ounces of olives than 2 the 2-ounce can; this means that per ounce, the 34-ounce can is about the same price as the 2-ounce can. So neither can is really a better deal than the other in terms of price per ounce.

10 6. Suppose my salary is cut by 7%, but then the following year I receive a raise of 20%. Am I making more money than before? Explain. Solution: No. This is an example of a shifing reference value problem. new salary = (old salary - 7% of old salary)+ 20% of (old salary - 7% of old salary) = (old salary -.7(old salary))+.20(old salary -.7(old salary)) = 0.83(old salary)+.20(0.83(old salary)) = 0.996(old salary) So my new salary is 99.6% of what I made before I m actually making less! 7. If you have a 20% off coupon on an item that is already 40% off, my friend thinks this means the item is 60% off. Is he correct? Explain. Solution: No. Since the item is 40% off, its price is 00% 40% = 60% of its old value. Careful consideration gives new price = 60% of old price - 20% of 60% of old price =.6(old price).2(.6)(old price) = (.6.2.6)(old price) = 0.48(old price) = 48% of the old price So the item s new price is 48% of its old price. In other words, the item is now 00% 48% = 52% off. If this is difficult to see, try calculating the amount off with a specific price (like $00).