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U of U MATH 1030 - Approaches to Problem Solving

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Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section ASection A.1: You Can’t Add Apples and OrangesDefinition of unitsThe units of a quantity describe what is measured or counted.We cannot add or subtract numbers with different units, but we can multiply and divide quantities withdifferent units.There are four different operations that we are allowed to do with d ifferent units.(1) DivisionEx.1If you drove 100 miles in two hours, your average speed was the distance traveled (= 100 mi) d ivid ed bythe time it took (= 2 hr):We read this answer as 50 “miles per hour”.Note that the word per means “divided by”.(2) Raising to second powerEx.2Find the area of a room 12 feet long and 10 feet wide.We say that the area is “120 square fe et”.The word square means something raised to the second power.1Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section A(3) Raising to third powerEx.3Find the volume of a box 6 inches wide, 4 inches deep and 10 inches long.We say that the volume is “240 cubic inches”.The word cubic means something raised to the third power.(4) MultiplicationEx.4Find the energy used by a 0.5-kilowatt light bulb for 6 hours.We say that the energy is “3 kilowatt-hours”.2Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AIdentifying UnitsIdentifying units in a problemIn order to understand what a problem is about, the first thing to do is to look at the units involved in theproblem.Ex.5Identify the units in the price for gasoline found by dividing its total cost in dollars by the number ofgallons of gas.Ex.6Identify the units in the area of the circle πr2, where r is the radius of the circle measured in centime-ters.Ex.7Identify the units in the volume found by multiplying an area measured in acres by a depth measured infeet.Section A.2: Unit ConversionsUnit conversionsIt is very important to understand how we can convert one unit to another one.Ex.8The statement 12 in. = 1 ft is an example of conversion factor. We can write this conversion factors inthree equivalent ways:3Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.9 Feet to Inches.Convert a distance of 5 feet into inches.Ex.10 Inches to Feet.Convert a length of 102 inches to feet.We ca n use many conversion factors to get the answer of a problem.Ex.11 Chain of conversions.How many seconds are there in one day?Conversions with Units Raised to PowersEx.12Convert 1 yd2to ft2.4Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.13Convert 1 yd3to ft3.Ex.14You want to carpet a room that measures 10 feet by 12 feet, making an area of 120 square feet. But carpetis usually sold by the square yard rather than by the square foot. How many square yards of c arpet doyou need?Ex.15You are preparing a vegetable garden that is 40 feet long and 16 feet wide, and you need enough soil tofill it to a depth of 1 foot. The landscape supply store sells soil by the cubic yard. How much soil shouldyou order?Currency ConversionsCurrencyDifferent countries use different money, or currency.5Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.161 euro = $1.272.Ex.17$1 = 10.95 pesos.Ex.18 Price conversion.At a french department store, the price for a pair of jeans is 45 euros. What is the price in U.S. dol-lars?Ex.19 Buying Currency.You are on holiday in Mexico and need cash. How many pesos can you buy with 100 U.S. dollars? Assumethat the transaction d oes not involve any fees.6Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section ASection A.3: Problem Solving with UnitsWorking with units(1) Identify the units involved in the problem. Use the units to help you decide how to approach theproblem and what units to expect in the answer.(2) Perform any operations (addition, multiplication,...) on both the numbers and the associated units.Remember:• You cannot add or subtract numbers with different units (you cannot add apples and oranges),but you can combine d ifferent units through multiplication, division or raising to powers.• To make your work with units ea sier, replace division with multiplication by the reciprocal.(3) When you complete your calculations, make sure that your answer has the units you expected. If itdoesn’t, then there is a mistake.Ex.20A car is traveling 25 miles every half-hour. How fast is it going?Ex.21You are buying 30 km2of farm land at a cost of $12, 000 per km2. What is the total cost?7Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.22 Exam Check.You are a grader for a math course. An exam question reads “Robert purchased 5 pounds of apples ata price of 50 cents per pound. How much does he pay for the apples?”. On the paper you are gradinga student has written “505= 10. He paid 10 cents.”. Write a note to the student explaining what wentwrong.8Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.23 Gas Mileage.After a long day of dr iving, your destination is only 90 miles away. You know that your car gets 25 milesper gallon. How much gas do you need in your tank if you are to reach the destination without stopping?If you have a 12-gallon tank and the fuel gauge shows it is one-quarter full, will you make it?9Chapter 2: Approaches to Problem Solving Lecture notes Math 1030 Section AEx.24 Melting of the Ice Caps.Measurements of polar ice show that, if all Earth’s polar ice melted, about 25 million cubic kilometers ofwater would be added to the oceans, most of it coming from Antarctica. How much would sea level riseas a result? The total surface area of the Earth’s oceans is about 340 million square


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